user13375: Imagine a human being born in 1982 weighing a few pounds at birth and labeled 'Melinda' by her parents. This little human being grows up - as little human beings tend to do - and by 2020 we imagine her as an adult no longer called 'Melinda', but rather change her name to 'Abigail.' If we examine the bodies of Melinda and Abigail down to the atom and perform a thorough and exact accounting we find that Melinda and Abigail do not share even one atom in common. Their brains are composed of entirely different atoms. 
Is the rupa skandha of Melinda and Abigail the same or different? Has it been reborn? What was the manner of its rebirth? How did it occur? How many times did it occur between 1982 to 2020? Is the same true of the other skandhas... have they been reborn? Are they reborn due to 'identification' with an 'I' or due to physical laws or some combination? 
Did 'Melinda' die sometime between 1982 and 2020 and get utterly annihilated? Was Abigail born for the first time from scratch between 1982 and 2020 and just pop into existence from nothing?
How is it possible that beings are reborn from moment-to-moment? In what manner and to what extent? What did the Buddha teach?

user295: 
OP: Are the Skandha's reborn from moment-to-moment?

Yes.

Viññāṇapaccayā nāma-rūpaṃ, “dependent on consciousness arises mentality-materiality.” The term nāma here stands for the mental states (cetasika), in other words, the three mental groups: namely, feeling (vedanākkhandha), perception (saññākkhandha), and volitional or mental formations (saṅkhārakkhandha).
The so called “being” (satta, Skt. sattva) is composed of five aggregates or groups (pañcakkhandha); namely, physical body, feeling, perception, volitional formations, and consciousness (rūpa, vedanā, saññā, saṅkhārā and viññāṇa). If consciousness is taken as the mind, then feeling, perceptions and volitional formations are the concomitants or factors of that mind. Now when we say dependent on consciousness arises nāma-rūpa, mentality-materiality, materiality means the physical body, its organs, faculties, and functions. Mentality means the factors of the mind mentioned above. In other words, viññāṇa-paccayā nāma-rūpaṃ means dependent on consciousness arise the three mental concomitants (feeling, perception, and volitional formations) that compose mentality, along with the conascent material body in its first embryonic stage.
Consciousness and its factors (citta-cetasika) are always interrelated and interdependent. Consciousness cannot arise and function independently of its factors, nor can the factors arise and function without consciousness. They arise simultaneously (sahajāta-paccaya) and have no independent existence.
The function of viññāṇa, consciousness, is varied. The third factor of the chain is made known to us as viññāṇa; now here again we hear of a sixth base, manāyatana, which is identical with consciousness. But here by manāyatana different types of consciousness are meant. It should be borne in mind that consciousness is not something that is permanent and everlasting. It undergoes change, not remaining the same for two consecutive moments; it comes into being and immediately passes away yielding place to a new consciousness. “These mental phenomena are, as it were, only the different aspects of those units of consciousness which like lightning every moment flash up and immediately thereafter disappear forever.”

Source: Dependent Origination (Paṭicca Samuppāda) by Piyadassi Thera

OP: Is the rupa skhanda of Melinda and Abigail the same or different?

This is neither the same nor different. There is not core which is transmitted but each series of Skandas are a product of same DO process.

OP: Has it been reborn?

Yes. They die and recreated as the wheel of DO roles forward.

There are three kinds of death: death as
  cutting off, momentary death, and conventional death. Death as cutting off belongs
  to those whose cankers are exhausted (and are Arahants). Momentary death is
  that of each consciousness of the cognitive series beginning with life-continuum
  consciousness, which arise each immediately on the cessation of the one preceding.
  Conventional death is that of all (so-called) living beings. Mine will be conventional
  death.

Introduction to The Path of Purification Visuddhimagga

OP: What was the manner of its rebirth? How did it occur?

Past conditioning gives rise to the consciousness, mind and matter and 6 sense bases for the monetary experience. The craving towards the feelings is the fuel which keeps the cycle going. 

OP: How many times did it occur between 1982 to 2020?

Rūpa also arises and dissolves at a tremendous rate of more
than 58 billion times per second. So for such a long period, it is difficult to say.

The Life-Time of Citta
Citta (consciousness) arises and dissolves in a person at a
  tremendous rate of more than a thousand billion times per eye-wink
  and there are about 250 eye-winks in a second. So, the life-time of a
  citta is less than one-thousand billionth of a second.
The life-time or duration of a citta is measured by three short
  instants, characterising the distinct features in the arising and passing
  of the citta. These are: (i) uppāda- the arising instant, (ii) ṭhiti- the
  presence or existing instant and (iii) bhanga- the dissolving instant.
  These three short instants (khaṇas) are said to be “one moment of
  consciousness” or “one conscious moment” (cittakkhaṇa). So, the lifetime of a citta is equal to the three short instants of arising, existing
  and dissolving of citta, i.e. it is equal to one conscious moment
  (cittakkhaṇa).
The Life-Time of Matter-Rūpa
The life-time of matter or Rūpa is 17 times longer than that of
  citta. So, we can say that the life-time of rūpa is equal to 17
  cittakkhaṇas, or 17 conscious moments, or 51 short instants (17X3 =
  51); as there are 3 short instants in a moment of consciousness.
Thus, rūpa also arises and dissolves at a tremendous rate of more
  than 58 billion times per second. The difference between citta and
  rūpa is as follows: citta arise one after another, whereas rūpa arise by
  manifesting as thousands of units in a small instant and it goes on
  constantly arising at every small instant in time. Therefore, rūpa may
  accumulate to become large masses that are visible to the naked eye,
  whereas the fleeting stream of consciousnesses is invisible to the
  naked eye. 

Process of Consciousness and Matter by Rewata Dhamma

OP: Is the same true of the other skhandas... have they been reborn? 

Yes.

OP: Are they reborn due to 'identification' with an 'I' or due to physical laws or some combination?

This is due to conditioning fueled by craving. Self-identification leads to craving. See this, this and this answer.

user8157: 
Is the rupa skhanda of Melinda and Abigail the same or different?

The teachings say the khandhas alter, change (SN 22.1), are impermanent (SN 22.59). 

Has it been reborn?

No. The Pali words translated as 'reborn' are associated with "kamma" or "volition". The change & alteration of the rupa khandha over the lifetime is not related to volition (apart from volitionally produced change, such as from eating too much or having plastic surgery). 

What was the manner of its rebirth?

Its not reborn. Only identity towards the rupa khandha is 'reborn'. 

Are they reborn due to 'identification' with an 'I' or due to physical
  laws or some combination?

Reborn due to 'identification' with an 'I' 

Did 'Melinda' die sometime between 1982 and 2020 and get utterly annihilated? 

Yes, it seems so. In the Pali suttas (DN 1; Iti 49; etc) the term "annihilation" refers to the annihilation of a "self-view" or "existent being".

Was Abigail born for the first time from scratch between 1982 and 2020 and just pop into existence from nothing?

Yes. But not from nothing. Abligail was born from ignorance & craving. 

How is it possible that beings are reborn from moment-to-moment? In
  what manner and to what extent? What did the Buddha teach?

SN 12.2 defines "jati" or "birth" as a "category of beings produced by the manifestation of their aggregates and the acquisition of their sense bases (of identification)". 
For example, Abigail craves to be attractive, manifests her aggregates by wearing make up and sex clothing and mental behaving in certain seductive ways. She identifies herself within the category of "sexy women". If men don't pay attention to her, sorrow arises. If other sexy women look more sexy than her, despair arises. Sorrow & despair arise with the "aging & death" of her fragile identity. Despite these setbacks, her repetition of such sexualising behaviour is "rebirthing" or "upapajjati". 
The word "upapatti/upapajjati" ("rebirth/reborn") appears not related to the word "jati" ("birth"). 'Upapajjati' is 'upa + pad + ya'; where the root 'pad' appears to mean 'to walk; to move' and the suffix 'upa' means 'close to' or 'near'. 'Jati' appears to be a noun from the verb root 'jan'; which appears to mean "bring forth'. 
Therefore, "jati" appears to mean "bring forth" and "upapajjati" appears to mean "to continue/be in a similar manner as before". 
For example, SN 12.10, includes both "jati" & "upapajjati", where the word "jayati" is the "verb" for the noun "jati", as follows: 

Mendicants, before my awakening — when I was still unawakened but
  intent on awakening— I thought:
Pubbeva me, bhikkhave, sambodhā anabhisambuddhassa bodhisattasseva
  sato etadahosi:
‘Alas, this world has fallen into trouble. It is born, grows old, dies,
  passes away, and is reborn [continues in the same manner as before]...
‘kicchaṃ vatāyaṃ loko āpanno jāyati ca jīyati ca mīyati ca cavati ca
  upapajjati ca.


user43: Imagine a camp fire that started in 1982. The original fuel has burned down but new fuel is regularly added. The overall shape and configuration of the fire in 2020 is somewhat reminiscent of the original 1982 fire but of course none of the material is the same.
Now if we were to analytically separate the fire into several categories (skandhas) we would find wood, ash, heat, light, oxygen, and smoke - all playing their roles in the maintenance (upadana) of fire.
Is it the same fire staying or is the fire "reborn" moment to moment? It's neither and both, these are just concepts to explain the working of causal continuity.
One important mistake in the worldview underlying your original question:  it assumes that "the fire" starts when a new human is born. But this is not so. The fire has been continuously burning through generations. "The beginning point is not evident". A new human is just another tongue of flame.

13375: <p>Imagine a human being born in 1982 weighing a few pounds at birth and labeled 'Melinda' by her parents. This little human being grows up - as little human beings tend to do - and by 2020 we imagine her as an adult no longer called 'Melinda', but rather change her name to 'Abigail.' If we examine the bodies of Melinda and Abigail down to the atom and perform a thorough and exact accounting we find that Melinda and Abigail do not share even one atom in common. Their brains are composed of entirely different atoms. </p>

<p>Is the rupa skandha of Melinda and Abigail the same or different? Has it been reborn? What was the manner of its rebirth? How did it occur? How many times did it occur between 1982 to 2020? Is the same true of the other skandhas... have they been reborn? Are they reborn due to 'identification' with an 'I' or due to physical laws or some combination? </p>

<p>Did 'Melinda' <em>die</em> sometime between 1982 and 2020 and get utterly annihilated? Was Abigail <em>born</em> for the first time from scratch between 1982 and 2020 and just pop into existence from nothing?</p>

<p>How is it possible that beings are reborn from moment-to-moment? In what manner and to what extent? What did the Buddha teach?</p>
 295: <blockquote>
  <p><strong>OP: Are the Skandha's reborn from moment-to-moment?</strong></p>
</blockquote>

<p>Yes.</p>

<blockquote>
  <p>Viññāṇapaccayā nāma-rūpaṃ, “dependent on consciousness arises mentality-materiality.” The term nāma here stands for the mental states (cetasika), in other words, the three mental groups: namely, feeling (vedanākkhandha), perception (saññākkhandha), and volitional or mental formations (saṅkhārakkhandha).</p>
  
  <p>The so called “being” (satta, Skt. sattva) is composed of five aggregates or groups (pañcakkhandha); namely, physical body, feeling, perception, volitional formations, and consciousness (rūpa, vedanā, saññā, saṅkhārā and viññāṇa). If consciousness is taken as the mind, then feeling, perceptions and volitional formations are the concomitants or factors of that mind. Now when we say dependent on consciousness arises nāma-rūpa, mentality-materiality, materiality means the physical body, its organs, faculties, and functions. Mentality means the factors of the mind mentioned above. In other words, viññāṇa-paccayā nāma-rūpaṃ means dependent on consciousness arise the three mental concomitants (feeling, perception, and volitional formations) that compose mentality, along with the conascent material body in its first embryonic stage.</p>
  
  <p>Consciousness and its factors (citta-cetasika) are always interrelated and interdependent. Consciousness cannot arise and function independently of its factors, nor can the factors arise and function without consciousness. They arise simultaneously (sahajāta-paccaya) and have no independent existence.</p>
  
  <p>The function of viññāṇa, consciousness, is varied. The third factor of the chain is made known to us as viññāṇa; now here again we hear of a sixth base, manāyatana, which is identical with consciousness. But here by manāyatana different types of consciousness are meant. It should be borne in mind that consciousness is not something that is permanent and everlasting. It undergoes change, not remaining the same for two consecutive moments; it comes into being and immediately passes away yielding place to a new consciousness. “<strong>These mental phenomena are, as it were, only the different aspects of those units of consciousness which like lightning every moment flash up and immediately thereafter disappear forever.</strong>”</p>
</blockquote>

<p>Source: <a href="https://www.bps.lk/olib/wh/wh015_Piyadassi_Dependent-Origination.html" rel="nofollow noreferrer">Dependent Origination (Paṭicca Samuppāda)</a> by Piyadassi Thera</p>

<blockquote>
  <p><strong>OP: Is the rupa skhanda of Melinda and Abigail the same or different?</strong></p>
</blockquote>

<p>This is neither the same nor different. There is not core which is transmitted but each series of Skandas are a product of same DO process.</p>

<blockquote>
  <p><strong>OP: Has it been reborn?</strong></p>
</blockquote>

<p>Yes. They die and recreated as the wheel of DO roles forward.</p>

<blockquote>
  <p>There are three kinds of death: death as
  cutting off, momentary death, and conventional death. Death as cutting off belongs
  to those whose cankers are exhausted (and are Arahants). Momentary death is
  that of each consciousness of the cognitive series beginning with life-continuum
  consciousness, which arise each immediately on the cessation of the one preceding.
  Conventional death is that of all (so-called) living beings. Mine will be conventional
  death.</p>
</blockquote>

<p><a href="https://www.accesstoinsight.org/lib/authors/nanamoli/PathofPurification2011.pdf" rel="nofollow noreferrer">Introduction to The Path of Purification Visuddhimagga</a></p>

<blockquote>
  <p><strong>OP: What was the manner of its rebirth? How did it occur?</strong></p>
</blockquote>

<p>Past conditioning gives rise to the consciousness, mind and matter and 6 sense bases for the monetary experience. The craving towards the feelings is the fuel which keeps the cycle going. </p>

<blockquote>
  <p><strong>OP: How many times did it occur between 1982 to 2020?</strong></p>
</blockquote>

<p>Rūpa also arises and dissolves at a tremendous rate of more
than 58 billion times per second. So for such a long period, it is difficult to say.</p>

<blockquote>
  <p><strong>The Life-Time of Citta</strong></p>
  
  <p>Citta (consciousness) arises and dissolves in a person at a
  tremendous rate of more than a thousand billion times per eye-wink
  and there are about 250 eye-winks in a second. So, the life-time of a
  citta is less than one-thousand billionth of a second.</p>
  
  <p>The life-time or duration of a citta is measured by three short
  instants, characterising the distinct features in the arising and passing
  of the citta. These are: (i) uppāda- the arising instant, (ii) ṭhiti- the
  presence or existing instant and (iii) bhanga- the dissolving instant.
  These three short instants (khaṇas) are said to be “one moment of
  consciousness” or “one conscious moment” (cittakkhaṇa). So, the lifetime of a citta is equal to the three short instants of arising, existing
  and dissolving of citta, i.e. it is equal to one conscious moment
  (cittakkhaṇa).</p>
  
  <p><strong>The Life-Time of Matter-Rūpa</strong></p>
  
  <p>The life-time of matter or Rūpa is 17 times longer than that of
  citta. So, we can say that the life-time of rūpa is equal to 17
  cittakkhaṇas, or 17 conscious moments, or 51 short instants (17X3 =
  51); as there are 3 short instants in a moment of consciousness.</p>
  
  <p>Thus, rūpa also arises and dissolves at a tremendous rate of more
  than 58 billion times per second. The difference between citta and
  rūpa is as follows: citta arise one after another, whereas rūpa arise by
  manifesting as thousands of units in a small instant and it goes on
  constantly arising at every small instant in time. Therefore, rūpa may
  accumulate to become large masses that are visible to the naked eye,
  whereas the fleeting stream of consciousnesses is invisible to the
  naked eye. </p>
</blockquote>

<p><a href="http://host.pariyatti.org/treasures/Process_of_Consciousness_and_Matter.pdf" rel="nofollow noreferrer">Process of Consciousness and Matter</a> by Rewata Dhamma</p>

<blockquote>
  <p><strong>OP: Is the same true of the other skhandas... have they been reborn?</strong> </p>
</blockquote>

<p>Yes.</p>

<blockquote>
  <p><strong>OP: Are they reborn due to 'identification' with an 'I' or due to physical laws or some combination?</strong></p>
</blockquote>

<p>This is due to conditioning fueled by craving. Self-identification leads to craving. See <a href="https://buddhism.stackexchange.com/a/35119/295">this</a>, <a href="https://buddhism.stackexchange.com/a/34099/295">this</a> and <a href="https://buddhism.stackexchange.com/a/35047/295">this</a> answer.</p>
 8157: <blockquote>
  <p>Is the rupa skhanda of Melinda and Abigail the same or different?</p>
</blockquote>

<p>The teachings say the khandhas alter, change (SN 22.1), are impermanent (SN 22.59). </p>

<blockquote>
  <p>Has it been reborn?</p>
</blockquote>

<p>No. The Pali words translated as 'reborn' are associated with "kamma" or "volition". The change &amp; alteration of the rupa khandha over the lifetime is not related to volition (apart from volitionally produced change, such as from eating too much or having plastic surgery). </p>

<blockquote>
  <p>What was the manner of its rebirth?</p>
</blockquote>

<p>Its not reborn. Only identity towards the rupa khandha is 'reborn'. </p>

<blockquote>
  <p>Are they reborn due to 'identification' with an 'I' or due to physical
  laws or some combination?</p>
</blockquote>

<p>Reborn due to 'identification' with an 'I' </p>

<blockquote>
  <p>Did 'Melinda' die sometime between 1982 and 2020 and get utterly annihilated? </p>
</blockquote>

<p>Yes, it seems so. In the Pali suttas (DN 1; Iti 49; etc) the term "annihilation" refers to the annihilation of a "self-view" or "existent being".</p>

<blockquote>
  <p><em>Was Abigail born for the first time from scratch between 1982 and 2020 and just pop into existence from nothing?</em></p>
</blockquote>

<p>Yes. But not from nothing. Abligail was born from ignorance &amp; craving. </p>

<blockquote>
  <p>How is it possible that beings are reborn from moment-to-moment? In
  what manner and to what extent? What did the Buddha teach?</p>
</blockquote>

<p>SN 12.2 defines "<em>jati</em>" or "birth" as a "<em>category of beings produced by the manifestation of their aggregates and the acquisition of their sense bases (of identification)</em>". </p>

<p>For example, Abigail craves to be attractive, manifests her aggregates by wearing make up and sex clothing and mental behaving in certain seductive ways. She identifies herself within the category of "sexy women". If men don't pay attention to her, sorrow arises. If other sexy women look more sexy than her, despair arises. Sorrow &amp; despair arise with the "aging &amp; death" of her fragile identity. Despite these setbacks, her repetition of such sexualising behaviour is "rebirthing" or "upapajjati". </p>

<p>The word "upapatti/upapajjati" ("rebirth/reborn") appears not related to the word "jati" ("birth"). 'Upapajjati' is 'upa + pad + ya'; where the root 'pad' appears to mean 'to walk; to move' and the suffix 'upa' means 'close to' or 'near'. 'Jati' appears to be a noun from the verb root 'jan'; which appears to mean "bring forth'. </p>

<p>Therefore, "jati" appears to mean "bring forth" and "upapajjati" appears to mean "to continue/be in a similar manner as before". </p>

<p>For example, SN 12.10, includes both "jati" &amp; "upapajjati", where the word "jayati" is the "verb" for the noun "jati", as follows: </p>

<blockquote>
  <p><em>Mendicants, before my awakening — when I was still unawakened but
  intent on awakening— I thought:</em></p>
  
  <p><em>Pubbeva me, bhikkhave, sambodhā anabhisambuddhassa bodhisattasseva
  sato etadahosi:</em></p>
  
  <p><em>‘Alas, this world has fallen into trouble. It <strong>is</strong> <strong>born</strong>, grows old, dies,
  passes away, and <strong>is</strong> <strong>reborn [continues in the same manner as before]</strong>...</em></p>
  
  <p><em>‘kicchaṃ vatāyaṃ loko āpanno <strong>jāyati</strong> ca jīyati ca mīyati ca cavati ca
  <strong>upapajjati</strong> ca.</em></p>
</blockquote>
 43: <p>Imagine a camp fire that started in 1982. The original fuel has burned down but new fuel is regularly added. The overall shape and configuration of the fire in 2020 is somewhat reminiscent of the original 1982 fire but of course none of the material is the same.</p>

<p>Now if we were to analytically separate the fire into several categories (skandhas) we would find wood, ash, heat, light, oxygen, and smoke - all playing their roles in the maintenance (upadana) of fire.</p>

<p>Is it the same fire staying or is the fire "reborn" moment to moment? It's neither and both, these are just concepts to explain the working of causal continuity.</p>

<p>One important mistake in the worldview underlying your original question:  it assumes that "the fire" starts when a new human is born. But this is not so. The fire has been continuously burning through generations. "The beginning point is not evident". A new human is just another tongue of flame.</p>
