user14387: Didn't the Buddha do some kind of great compassion meditation  for a significant part of his daily routine while teaching? If so, what's the name of it and what did it involve?

user21202: Compassionate meditation aka (Karuna meditation) is a form of meditation to develop compassion for others and ourselves. Buddhism holds a significant amount of importance towards compassion. Practicing Loving Kindness, using Mantras, Mudras, Affirmations are ideal for this form of meditation.
Steps to be followed while considering this form of meditation are:

Find somewhere quiet where you will not be disturbed. Sit with a good and comfortable posture. Place your feet shoulder width apart. Sit with a straight but relaxed spine.
Close your eyes and spend sometime with mindful breathing. This helps in promoting relaxation and focus.
Bring to mind a person who has currently or is suffering. Begin with people for whom  you feel the most sympathy. Sincerity is everything in Karuna meditation. Therefore, start with someone you love.
Consider the struggles this person is facing.
Wish the individual freedom from suffering. Wish for them to be happier, healthier, more fortunate and more successful. Or if a person is ill you wish for them to get healthier!
If you feel any conflicting emotions as such, just observe them don't be judgemental about the situation.
Observe the feeling of compassion. Be mindful of it. How does the feeling feel in the body and mind?. Are there any obstacles in the way to genuine compassion? Compassion is a feeling. And feelings are a source of energy! Connect with the inner energy of compassion. And thats the gateway towards feeling like a Boddhisatva.
Repeat the script meditating on different people.
Finish with a wish to free all sentient beings from suffering.
Sometimes, we may not feel compassion for others and sometimes even for ourselves, if that is the case, we need to start off with forgiveness.

And thats where Meditation on forgiveness starts.

user19480: Great Compassion, Mahākaruṇā, is different than simple karuṇā, or compassion. It is more properly translated as “great responsiveness” and is the activity, or functioning, of Emptiness. Unlike karuṇā—compassion that must be developed—mahākaruṇā is the intrinsic naturing of all appearances, is Buddha-nature, and thus the Buddha naturally manifested this responsiveness in his every word and deed. Technically, this is not the result of any practice, but is, rather, the fruit of enlightenment. See:

The final attribute of emptiness to be mentioned is a quality peculiar
to the Buddhist analysis: responsiveness. It is the third and final
denominator in the list of categories or aspects by which emptiness
can be defined: essence, nature, responsiveness. It appears anomalous,
an attribute rather than a category. The third logical category is
function, or manifest function, and the attribute found in its stead
is responsiveness and its qualifier is all-pervasive. Viewed as a
functional attribute of inner space, total presence, and light, the
implication is that the dynamic, the intentionality, the purpose of
being is compassion, which is a synonym of responsiveness and
demonstrable as the responsive aspect of love. It is this compassion
that is coextensive with space, the buddha-heart pervading all beings.
Viewed as the potential form or manifestation of emptiness, the
implication appears to be that every vibration of body, speech, and
mind is a form of compassionate energy, nothing excluded. Consider the
distinction between responsiveness and compassion. In Dzogchen,
compassion is much more than the virtue of loving-kindness. Nor does
the word compassion in the Dzogchen context denote its English
etymological meaning, “suffering together” or “empathy,” although both
these meanings may be inferred. Essentially, compassion indicates an
open and receptive mind responding spontaneously to the exigencies of
an ever-changing field of vibration to sustain the optimal awareness
that serves self-and-others’ ultimate desire for liberation and
well-being. The conventional meaning of compassion denotes the latter,
active part of this definition, and, due to the accretions of
Christian connotation in the West, response is limited to specifically
virtuous activity. Responsiveness defines the origin and cause of
selfless activity that can encompass all manner of response. On this
nondual Dzogchen path, virtue is the effect, not the cause; the
ultimate compassionate response is whatever action optimizes presence
— loving-kindness is the automatic function of primal awareness. (Excerpt From: “The Flight of the Garuda,” by Keith Dowman, Wisdom Publications, 2003, pgs 37-38)

However, the meditation that Buddha continued throughout his life, the meditation practice that brought him to realization, is explained in the Surangama Sutra, and its results, in the Surangama Samadhi Sutra. In the Surangama Sutra it is explained by Avalokitasvara as being his practice, resulting in the perfection of mahākaruṇā.
Manjushri, in the same Sutra, explains why this practice is different than all others—it has to do with the support used—and asserts that this is the practice used by he himself, Avalokitasvara, the Buddha, and all Buddhas. He explains why this is so, in the case of all Buddhas—because there is no Dharma extant when a Buddha arises in the world, thus, something other than the Dharma is needed to reach full enlightenment since all other meditation supports are discontinuous. I recommend reading the sections in that Sutra: “The Method Of Avalokitasvara Bodhisattva,” and “Manjushri’s Summation”.
There is also a prophecy disseminated by Jamyang Khyentse Chokyi Lodro that prescribes the use of this practice to attenuate the suffering of our current time, and specifically during the years 2026 through 2032.
So, to directly answer your question, the name of the meditation is just Great Responsiveness Meditation, at least as used in the prophecy. A variant of it, called the Yoga of the Supreme Sound of the Four Elements was once used in Dzogchen practices from Tibetan Buddhism, and Bön, where is was lauded as the most efficacious practice leading to the manifestation of great responsiveness, i.e., enlightenment.

14387: <p>Didn't the Buddha do some kind of great compassion meditation  for a significant part of his daily routine while teaching? If so, what's the name of it and what did it involve?</p>
 21202: <p>Compassionate meditation aka (Karuna meditation) is a form of meditation to develop compassion for others and ourselves. Buddhism holds a significant amount of importance towards compassion. Practicing Loving Kindness, using Mantras, Mudras, Affirmations are ideal for this form of meditation.
Steps to be followed while considering this form of meditation are:</p>
<ol>
<li>Find somewhere quiet where you will not be disturbed. Sit with a good and comfortable posture. Place your feet shoulder width apart. Sit with a straight but relaxed spine.</li>
<li>Close your eyes and spend sometime with mindful breathing. This helps in promoting relaxation and focus.</li>
<li>Bring to mind a person who has currently or is suffering. Begin with people for whom  you feel the most sympathy. Sincerity is everything in Karuna meditation. Therefore, start with someone you love.</li>
<li>Consider the struggles this person is facing.</li>
<li>Wish the individual freedom from suffering. Wish for them to be happier, healthier, more fortunate and more successful. Or if a person is ill you wish for them to get healthier!</li>
<li>If you feel any conflicting emotions as such, just observe them don't be judgemental about the situation.</li>
<li>Observe the feeling of compassion. Be mindful of it. How does the feeling feel in the body and mind?. Are there any obstacles in the way to genuine compassion? Compassion is a feeling. And feelings are a source of energy! Connect with the inner energy of compassion. And thats the gateway towards feeling like a Boddhisatva.</li>
<li>Repeat the script meditating on different people.</li>
<li>Finish with a wish to free all sentient beings from suffering.</li>
<li>Sometimes, we may not feel compassion for others and sometimes even for ourselves, if that is the case, we need to start off with forgiveness.</li>
</ol>
<p>And thats where Meditation on forgiveness starts.</p>
 19480: <p>Great Compassion, Mahākaruṇā, is different than simple karuṇā, or compassion. It is more properly translated as “great responsiveness” and is the activity, or functioning, of Emptiness. Unlike karuṇā—compassion that must be developed—mahākaruṇā is the intrinsic naturing of all appearances, is Buddha-nature, and thus the Buddha naturally manifested this responsiveness in his every word and deed. Technically, this is not the result of any practice, but is, rather, the fruit of enlightenment. See:</p>
<blockquote>
<p>The final attribute of emptiness to be mentioned is a quality peculiar
to the Buddhist analysis: responsiveness. It is the third and final
denominator in the list of categories or aspects by which emptiness
can be defined: essence, nature, responsiveness. It appears anomalous,
an attribute rather than a category. The third logical category is
function, or manifest function, and the attribute found in its stead
is responsiveness and its qualifier is all-pervasive. Viewed as a
functional attribute of inner space, total presence, and light, the
implication is that the dynamic, the intentionality, the purpose of
being is compassion, which is a synonym of responsiveness and
demonstrable as the responsive aspect of love. It is this compassion
that is coextensive with space, the buddha-heart pervading all beings.
Viewed as the potential form or manifestation of emptiness, the
implication appears to be that every vibration of body, speech, and
mind is a form of compassionate energy, nothing excluded. Consider the
distinction between responsiveness and compassion. In Dzogchen,
compassion is much more than the virtue of loving-kindness. Nor does
the word compassion in the Dzogchen context denote its English
etymological meaning, “suffering together” or “empathy,” although both
these meanings may be inferred. Essentially, compassion indicates an
open and receptive mind responding spontaneously to the exigencies of
an ever-changing field of vibration to sustain the optimal awareness
that serves self-and-others’ ultimate desire for liberation and
well-being. The conventional meaning of compassion denotes the latter,
active part of this definition, and, due to the accretions of
Christian connotation in the West, response is limited to specifically
virtuous activity. Responsiveness defines the origin and cause of
selfless activity that can encompass all manner of response. On this
nondual Dzogchen path, virtue is the effect, not the cause; the
ultimate compassionate response is whatever action optimizes presence
— loving-kindness is the automatic function of primal awareness. (Excerpt From: “The Flight of the Garuda,” by Keith Dowman, Wisdom Publications, 2003, pgs 37-38)</p>
</blockquote>
<p>However, the meditation that Buddha continued throughout his life, the meditation practice that brought him to realization, is explained in the Surangama Sutra, and its results, in the Surangama Samadhi Sutra. In the Surangama Sutra it is explained by Avalokitasvara as being his practice, resulting in the perfection of mahākaruṇā.</p>
<p>Manjushri, in the same Sutra, explains why this practice is different than all others—it has to do with the support used—and asserts that this is the practice used by he himself, Avalokitasvara, the Buddha, and all Buddhas. He explains why this is so, in the case of all Buddhas—because there is no Dharma extant when a Buddha arises in the world, thus, something other than the Dharma is needed to reach full enlightenment since all other meditation supports are discontinuous. I recommend reading the sections in that Sutra: “<a href="https://medium.com/p/e0f3e124b72e" rel="nofollow noreferrer">The Method Of Avalokitasvara Bodhisattva</a>,” and “<a href="https://medium.com/p/ce0017f10d2b" rel="nofollow noreferrer">Manjushri’s Summation</a>”.</p>
<p>There is also a <a href="https://medium.com/p/c9e6e2c54f23" rel="nofollow noreferrer">prophecy disseminated</a> by Jamyang Khyentse Chokyi Lodro that prescribes the use of this practice to attenuate the suffering of our current time, and specifically during the years 2026 through 2032.</p>
<p>So, to directly answer your question, the name of the meditation is just Great Responsiveness Meditation, at least as used in the prophecy. A variant of it, called the Yoga of the Supreme Sound of the Four Elements was once used in Dzogchen practices from Tibetan Buddhism, and Bön, where is was lauded as the most efficacious practice leading to the manifestation of great responsiveness, i.e., enlightenment.</p>
