Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. If for some point $s \in S$ and some $n \ge 0$ we have $H^n(\mathscr{X}_s, \mathscr{F}_s) = 0$, does it then follow that $R^nf_*(\mathscr{F}) = 0$ in a neighborhood of $s$?

For schemes, the claim is part of the cohomology and base change formalism. I have looked at some references on similar material for algebraic stacks (Grothendieck existence, semicontinuity of cohomology...), but could not find (or overlooked) the above statement. A pointer to the literature would be very welcome.