If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and there exists a natural map from the k-th Chow group A_k(X) to the k-th graded part Gr_k K_0(X), just by mapping [V] to [O_V]. This is example 15.1.5 in Fulton's book.

This even becomes an isomorphism after tensoring with Q! That's Corollary 18.3.2 in Fulton's book.

All of the definitions surely make sense for DM-stacks. We've got Chow-groups and K_0 can be graded in the same way. I'm not sure whether the natural map above actually passes to rational equivalence, but I'll just assume this to be true.

Here's the question: Is the map still an isomorphism?