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The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...

Consider the cartesian square of schemes $$ \require{AMScd} \begin{CD} X' @>{g'}>> X \\ @V{f'}VV @VV{f}V \\ S' @>>{g}> S \end{CD} $$ and the base change map $$ \eta : ...