Let $R$ be a complete local ring (even of dimension one if it helps) and write it as limit of artinian rings $R_n$. Let $X\rightarrow S=Spec(R)$ be proper, finite type even smooth outside the maximal ideal of $R$. Define $X_n$ as the pullback on $Spec(R_n)$. Consider $F$ a constructible sheaf of $\mathbb{Z}$-modules (finitely generated, even torsion free if you like) on $X$. Is it true that $R^1f_{et,*}F_{X_n}=R^1f_{et,*}F_{X_0}=R^1f_{et,*}F_{X}$? if not, are there examples with the relative dimension of the fibers $\geq 1$ where this holds ?
