I want to use some results true for non-singular varieties on smooth Deligne-Mumford stacks.
1 -- Concrete example: equivariant pullback for cohomology and K-theory of singular spaces that we could also think of as smooth Deligne-Mumford stacks -- like various moduli spaces. (I'm looking at Chriss and Ginzburg for equivariant K-theory of varieties, and I looked at Edidin and Graham's Equivariant Intersection Theory briefly for Chow groups.) K-theoretic pullback for singular spaces is messy. Can I say these spaces are smooth as DM stacks and use results for nonsingular varieties?
2 -- Meta-question: how does one go about translating results of any kind, but particularly K-theoretic results for nonsingular schemes, to results for smooth Deligne-Mumford stacks? Under what conditions do results carry over? I know this is a very big question; is there a concise answer?