let $f:X\to S$ be a proper morphism of schemes with $S=Spec(A)$ affine. Consider $F$ a constructible sheaf on $X$. I am interested to know for which ring $B$ with morphism $Spec(B)\to Spec(A)$ is it true that the morphism
$(f_{*}F)\otimes B \to H^0(X_B, F_B)$
is an isomorphism.
I am interested expecially in the case $Spec(B)\to Spec(A)$ étale, or $B=k(p)$ the residue field of a point $p\in Spec(A)$.
Thanks