Working with animals to bring carbon back into the Soil uses holistic management principles. Holistic management was largely developed by Allan Savory of the Savory Institute and has been successfully put into practice by many others.
The idea of holistic management is to learn from nature and mimic the way great herding animals have lived. They grouped together, grazed and moved, never staying too long in one place. Ranchers move large numbers of animals on smaller confined areas of pasture for shorter times. These defined areas are called paddocks and this is referred to as rotational grazing. It allows the animals to enjoy healthy nutritious plants, trample plant material into the ground, brake up hard ground, and leave microbial active droppings. The combination of broken top soil, plant material and droppings with microbes adds carbon to the soil and improves soil health. Plants grow better and water holding capacity is improved. It's regenerative.
The effects of well-managed animals on pasture are dramatic!
Grasslands cover a large part of the surface of the earth. Holistic management can potentially restore carbon to soils on a large scale and substantially contribute to mitigating the negative effects of climate change.
Examples - Land Restoration with Holistic Management by Seth Itzkan, Planet Tech Associates
More Examples - Holistic Management Comparison Pictures by Sheldon Frith
Soil Carbon Cowboys by Peter Byck (vimeo 12:22 min)
An Introduction to Allan Savory:
How to Fight Desertification and Reverse Climate Change | TED (22 min)