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EDIT: Under amenable circumstances when the left adjoint $\pi^*$ is exact, you can get a sufficient condition for your isomorphism to hold. (I'm not sure what kind of topology you want to work with, or whether you're working with simply sheaves of abelian groups or sheaves of $\mathcal{O}$-modules or quasicoherent modules or...) In this case, $\pi_*$ preserves injectives and so there's a Grothendieck spectral sequence $$H^p(X, {\mathbb R}^q \pi_* F) \Rightarrow H^{p+q}(\mathcal{X},F)$$ and so a sufficient condition is for the higher direct image functors $\mathbb{R}^q \pi_* F$ to vanish for q > 0. These are the sheaves associated to the presheaves $U \mapsto H^q(\pi^{-1} U, F)$, and often their stalks are related to the cohomology of the fibers of $\pi$ - which are, in this case of a DM-stack, pretty much the group cohomology of the stabilizers.