Tonglen, which means taking and sending, is a meditation practice for opening to our innate compassion that is most associated with the Tibetan tradition of Buddhism. Pema Chodron provides a brief introduction to the practice that you can find here: How to Practice Tonglen
Dekila Chungyalpa was born in the Himalayan land of Sikkim and is now Director of the Loka Initiative at the Center for Healthy Minds at the University of Wisconsin. She practices a powerful adaptation of tonglen focused on a compassionate response to ecoanxiety and climate despair. You can find the guided meditation that she shares here: A Meditation Practice for Environmental Despair You can find a video of an expanded version of her guided meditation here: Giving Gratitude and Receiving Resilience.
There are significant differences between empathy and compassion that help understand why empathy can lead to burnout, but the practice of compassion rarely does. This article by Matthieu Ricard explains the distinction and the neuroscience research that documented the difference between the two: The Differences and the Relationship Between Empathy and Compassion.
The Global Compassion Coalition is an umbrella organization of many groups that are working to increase the level of compassion in the world. Rick Hanson, author of Buddha’s Brain and several other books is the main organizer of the coalition: Global Compassion Coalition.