The result you want is this.

Let $f : \mathscr{X} \to S$ be a proper tame DM stack with $S$ a scheme, and $g : S' \to S$ any morphism of schemes. Let $\mathscr{F}$ be a torsion sheaf on $\mathscr{X}$. Then the natural base change morphism $$g^\ast R^i f_\ast \mathscr{F} \to R^if'_\ast g'^\ast \mathscr{F}$$ is an isomorphism.

For a proof, see Theorem A.0.8 of Abramovich-Corti-Vistoli. They also provide a description of the stalks of the higher pushforwards, which is something we discussed yesterday.