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 it is sufficient to let $X,Y$ be schemes.
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**Edit**: Will Sawin pointed out that the question as stated was obviously false, I've edited it to remove that false statement.