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 $q\gg 0$?

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 $N(n)$.