Consider a morphism of schemes $f:X\rightarrow Y$ and an abelian étale sheaf $\mathcal{F}$ on $X$. Then the higher direct image is the sheafification of the presheaf on the étale site of $X$ defined by $$U\mapsto H^p_{ét}(X\times_Y U,\mathcal{F}_U).$$ My question is if there are good conditions on this setup and $U$ such that $R^pf_*\mathcal{F}(U)=H^p(X\times_Y U,\mathcal{F}_U).$ The reason why I hope for such a condition to exists comes from the coherent sheaf cohomology. Namely, when $Y$ is affine and $X\rightarrow \text{Spec}(A)$ is quasi-compact and quasi-separated, then for any open (actually flat suffices) $\text{Spec}B\rightarrow \text{Spec}(A)$, then by flat base change we have $H^p(X_B,\mathcal{F}_B)\cong H^p(X,\mathcal{F})\otimes_A B$ and since $R^pf_*\mathcal{F}=H^p(X,\mathcal{F})^{\sim}(\text{Spec}(B))=H^p(X,\mathcal{F})\otimes_A B$, we get that in this case the higher direct image functor actually computes the cohomology of the fiber. Hence it seems not quite hopeless that this could be true. Considering how important the base change theorem was in the above argument, I'd suspect we would need to assume $X$ to be proper.
Have a nice day!