user-1: As I understand the Third Noble Truth, the cessation of dukkha is brought about by ending craving. I can see how, for example, practicing renunciation, or seeing no self, assist us in achieving this, but how do ethics and compassion practices like lovingkindness lead us to achieving the end of craving? 

user111: Sila(morality), Samadhi(concentration), Panna(wisdom) is the way to cessation of Dukkha. Sila disciplines bodily and verbal actions. Without that discipline you cannot discipline the mind, without disciplining the mind, you cannot cultivate wisdom.

user106: There is a strong aspect in the training by the Buddha to achieve some self-analysis, some self-liberation.     
But there is also another one, not often discussed but it occurs in some suttas (don't have the actual reference at the moment). This might be called a social one which introduces something like "reciprocity" or "echo": 
If you've taken the vow that you won't kill - and the people around you can trust you, then the people around you need not fear getting killed, they can themselves feel more free and secure, and this is not only a good athmosphere for them but shall echo back to yourself and your freedom to move between them - and in general for the common social environment.      
If you've taken the vow that you won't take what is not given - then the people around you need not fear that their things shall be taken away. When they know you are of that type you might walk through the fields (if it is a rural area) with much less things to be taken care of - so this again shall echo back to you - and in general for the common social environment (I really loved such a free- and trusting feeling when I wandered through the south of France, it was really new to me grown up in urban Germany...)
Think about the other vows - it should not be too difficult to image analoguous examples.           
So, your keeping-the-silas and generating-trust-in-this improves an environment of freedom and - one shall even find out- of generosity and other good mental states and routines, thus reducing dukkha on both sides in such reciprocal process.                   

(If I can remember the sutta I've read on this I'll add the reference later)

user43: Dukkha comes not just from craving, but from unsatisfied craving, in other words from inner conflict between "is" and "should".
Sila, Prajna, and Samadhi are three methods to remove and prevent that conflict: coarse, middling, and subtle conflict.
Sila helps remove and prevent coarse behavioral conflict, like obvious conflicted states of Obsession, Negativity, Egoism, - and their real life results, which manifest as various kinds of conflict. 
Prajna helps remove and prevent middling intellectual conflict, through analysis of dharmas, understanding how things work, understanding Three Marks of Existence, and Emptiness, understanding the limits of mind's models and perspectives.
Samadhi helps remove and prevent subtle emotional conflict, like states of negativity, unsatisfaction, craving for a different life etc.
So to answer your question, it helps by reducing, removing and preventing conflict between "is" and "should" which manifests as Dukkha. 

user471: Kimattha Sutta (AN 11.1) explains the relationship of skillful virtues (sila) to Nibbana:

Ven. Ananda asked: “What is the purpose of skillful virtues? What is their reward?”
The Buddha's answer:
“Skillful virtues have freedom from remorse as their purpose, Ananda,
  and freedom from remorse as their reward.”
“And what is the purpose of freedom from remorse? What is its reward?”
“Freedom from remorse has joy as its purpose, joy as its reward.”
“And what is the purpose of joy? What is its reward?”
“Joy has rapture as its purpose, rapture as its reward.”
“And what is the purpose of rapture? What is its reward?”
“Rapture has serenity as its purpose, serenity as its reward.”
“And what is the purpose of serenity? What is its reward?”
“Serenity has pleasure as its purpose, pleasure as its reward.”
“And what is the purpose of pleasure? What is its reward?”
“Pleasure has concentration as its purpose, concentration as its
  reward.”
“And what is the purpose of concentration? What is its reward?”
“Concentration has knowledge & vision of things as they actually are
  as its purpose, knowledge & vision of things as they actually are as
  its reward.”
“And what is the purpose of knowledge & vision of things as they
  actually are? What is its reward?”
“Knowledge & vision of things as they actually are has disenchantment
  as its purpose, disenchantment as its reward.”
“And what is the purpose of disenchantment? What is its reward?”
“Disenchantment has dispassion as its purpose, dispassion as its
  reward.”
“And what is the purpose of dispassion? What is its reward?”
“Dispassion has knowledge & vision of release as its purpose,
  knowledge & vision of release as its reward.


-1: <p>As I understand the Third Noble Truth, the cessation of dukkha is brought about by ending craving. I can see how, for example, practicing renunciation, or seeing no self, assist us in achieving this, but how do ethics and compassion practices like lovingkindness lead us to achieving the end of craving? </p>
 111: <p>Sila(morality), Samadhi(concentration), Panna(wisdom) is the way to cessation of Dukkha. Sila disciplines bodily and verbal actions. Without that discipline you cannot discipline the mind, without disciplining the mind, you cannot cultivate wisdom.</p>
 106: <p>There is a strong aspect in the training by the Buddha to achieve some self-analysis, some self-liberation.     </p>

<p>But there is also another one, not often discussed but it occurs in some suttas (don't have the actual reference at the moment). This might be called a social one which introduces something like "reciprocity" or "echo": </p>

<p>If you've taken the vow that you won't kill - and the people around you can trust you, then the people around you need not fear getting killed, they can themselves feel more free and secure, and this is not only a good athmosphere for them but shall echo back to yourself and your freedom to move between them - and in general for the common social environment.      </p>

<p>If you've taken the vow that you won't take what is not given - then the people around you need not fear that their things shall be taken away. When they know you are of that type you might walk through the fields (if it is a rural area) with much less things to be taken care of - so this again shall echo back to you - and in general for the common social environment (I really loved such a free- and trusting feeling when I wandered through the south of France, it was really new to me grown up in urban Germany...)</p>

<p>Think about the other vows - it should not be too difficult to image analoguous examples.           </p>

<p>So, your keeping-the-silas and generating-trust-in-this improves an environment of freedom and - one shall even find out- of generosity and other good mental states and routines, thus reducing dukkha on both sides in such reciprocal process.                   </p>

<p><hr>
<em>(If I can remember the sutta I've read on this I'll add the reference later)</em></p>
 43: <p>Dukkha comes not just from craving, but from <em>unsatisfied</em> craving, in other words from inner conflict between "is" and "should".</p>

<p>Sila, Prajna, and Samadhi are three methods to remove and prevent that conflict: coarse, middling, and subtle conflict.</p>

<p>Sila helps remove and prevent coarse behavioral conflict, like obvious conflicted states of Obsession, Negativity, Egoism, - and their real life results, which manifest as various kinds of conflict. </p>

<p>Prajna helps remove and prevent middling intellectual conflict, through analysis of dharmas, understanding how things work, understanding Three Marks of Existence, and Emptiness, understanding the limits of mind's models and perspectives.</p>

<p>Samadhi helps remove and prevent subtle emotional conflict, like states of negativity, unsatisfaction, craving for a different life etc.</p>

<p>So to answer your question, it helps by reducing, removing and preventing conflict between "is" and "should" which manifests as Dukkha. </p>
 471: <p><a href="https://www.accesstoinsight.org/tipitaka/an/an11/an11.001.than.html" rel="nofollow noreferrer">Kimattha Sutta (AN 11.1)</a> explains the relationship of skillful virtues (sila) to Nibbana:</p>

<blockquote>
  <p>Ven. Ananda asked: “What is the purpose of skillful virtues? What is their reward?”</p>
  
  <p>The Buddha's answer:</p>
  
  <p>“Skillful virtues have freedom from remorse as their purpose, Ananda,
  and freedom from remorse as their reward.”</p>
  
  <p>“And what is the purpose of freedom from remorse? What is its reward?”</p>
  
  <p>“Freedom from remorse has joy as its purpose, joy as its reward.”</p>
  
  <p>“And what is the purpose of joy? What is its reward?”</p>
  
  <p>“Joy has rapture as its purpose, rapture as its reward.”</p>
  
  <p>“And what is the purpose of rapture? What is its reward?”</p>
  
  <p>“Rapture has serenity as its purpose, serenity as its reward.”</p>
  
  <p>“And what is the purpose of serenity? What is its reward?”</p>
  
  <p>“Serenity has pleasure as its purpose, pleasure as its reward.”</p>
  
  <p>“And what is the purpose of pleasure? What is its reward?”</p>
  
  <p>“Pleasure has concentration as its purpose, concentration as its
  reward.”</p>
  
  <p>“And what is the purpose of concentration? What is its reward?”</p>
  
  <p>“Concentration has knowledge &amp; vision of things as they actually are
  as its purpose, knowledge &amp; vision of things as they actually are as
  its reward.”</p>
  
  <p>“And what is the purpose of knowledge &amp; vision of things as they
  actually are? What is its reward?”</p>
  
  <p>“Knowledge &amp; vision of things as they actually are has disenchantment
  as its purpose, disenchantment as its reward.”</p>
  
  <p>“And what is the purpose of disenchantment? What is its reward?”</p>
  
  <p>“Disenchantment has dispassion as its purpose, dispassion as its
  reward.”</p>
  
  <p>“And what is the purpose of dispassion? What is its reward?”</p>
  
  <p>“Dispassion has knowledge &amp; vision of release as its purpose,
  knowledge &amp; vision of release as its reward.</p>
</blockquote>
