Let $f:X\to Y$ be a representable map of Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf_*\mathbf{Q}_\ell=0$ for all $q>2n$ (or at least $q>N$ for some $N$)?

Note: by taking atlases, I think the we can just let $X,Y$ be schemes.

Of course the natural thing is to apply base change to $\require{AMScd}$ \begin{CD} F @> \overline{f} >> \text{pt}\\ @V V \overline{i}V @VV i V\\ X @> f> > Y \end{CD} but that gives $i^!f_*\mathbf{Q}_\ell=\overline{f}_*\overline{i}^!\mathbf{Q}_\ell$, which is not quite right. At least to conclude we would need to know that for all dimension $n$ schemes $F$, sheaves of the form $\overline{i}^!\mathbf{Q}_\ell$ have the degree of their cohomology bounded by $N(n)$.