Let $f: X \to S$ be a proper morphism of schemes and $\mathscr F$ a torsion abelian sheaf on $X$. Then, one version of the base change theorem in Etale Cohomology says, for a geometric point $s \in S$: $$(R^pf_*\mathscr F)_s = H^p(X_s, \mathscr F|_{X_s})$$ where $X_s$ is the fiber of $X$ at $s$ and $\mathscr G_s$ refers to the stalk of the sheaf at $s$.
In SGA 4 1/2, there is a remark to the effect that we can reduce the above theorem to the case where $S$ is a strictly henselian local ring. I do not see how this follows, even in the case where $p=0$.
As far as I can tell, we would have to show that the base change morphism for $\mathscr O_{S,\overline{s}} \to S$ is an isomorphism where $\mathscr O_{S,\overline{s}} = \varprojlim \mathscr O_U$ as the limit ranges over etale neighbourhoods $U$ of $S$. The base change morphism is clearly an isomorphism for any etale $U\to S$ but I don't see how to deal with the ivnerse limit.
SGA only spends a couple of lines on this so perhaps I am just missing some easy trick?
Here are some references: SP 03Q9; Fu, Etale Cohomology Theory, 5.9.5; SGA 4, Exp. VII, Section 5; Milne, Etale Cohomology, Theorem III.1.15.
If I understand correctly, the point is that (quasi-compact, quasi-separated) etale morphisms over a projective limit of schemes $X = \varprojlim X_{i}$ (with affine transition morphisms) can be approximated by an etale morphism over some $X_{i}$ as mentioned in SGA 4 1/2, Exp. 1, 2.3.3.