Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space.

What are some general results on the relationship between $H^i_{\mathrm{et}}(\mathscr{X}, \underline{\mathbf{Z}_\ell})$ and $H^i_{\mathrm{et}}(X, \underline{\mathbf{Z}_\ell})$? By the usual spectral sequence argument, I guess I'm asking for what some general results are about the pushforwards $R^i \pi_* \underline{\mathbf{Z}_\ell}$. If I'm not mistaken, proper base change should tell us that these are (constructible? lcc?) $\ell$-adic sheaves with stalks $H^i_{\mathrm{et}}(\mathscr{X}_x, \underline{\mathbf{Z}_\ell})$.

I'm happy for results that work in significantly less generality, or to know what some interesting conditions on $\mathscr{X}$ are which make this question easier. Conversely, I'd love to hear something that works when $\underline{\mathbf{Z}_\ell}$ is replaced with some other (lcc etc.) $\ell$-adic sheaf.

My motivation for this question came from the case where $\mathscr{X} = [Y/G]$ for $Y$ a smooth projective variety (even a hypersurface) over $\mathbf{C}$ and $G$ a finite cyclic group acting with non-discrete fixed points, and I wanted to compute torsion in the singular cohomology with $\mathbf{Z}$-coefficients.