Composting "in-place"?
Why do you want dirt? Grass under trees is common in yards, but uncommon in the wild. 

In the wild, things compost at the base of trees all the time. The best mulch for a tree is its own droppings + the droppings of the plants that like to grow under it. (e.g. ferns under cedars, salal under douglas fir). 

You can also just add a mulch around the tree. It holds in moisture and protects the soil. My favorite is shavings from a planer at the local wooden boat school. Peeing on the mulch provides nitrogen that helps break down the mulch and release its nutrients in to the soil, which the tree will like.