I am reading M. Artin's treatment of the proper base change theorem for étale cohomology in his "Théorèms de représentabilité pur les espaces algébriques", and I have trouble understanding the following remark on page 222:

If $f:X\rightarrow S$ and $g:S'\rightarrow S$ are morphisms of algebraic spaces (or schemes, if you prefer), and if $f':X'\rightarrow S'$, $g':X'\rightarrow X$ denote the base changes of $f$ and $g$, then one can construct for any abelian sheaf $F$ on the big étale site of $X$ the base change morphism
$g^*R^qf_*F\rightarrow R^q f'_*(g'^*F)$ (the higher direct images also computed on the big sites). If I understand correctly, Artin claims that if $F$, $R^q f_*F$ and $R^qf'_*(g'^*F)$ are representable on the big étale site of $X$, resp. $S$, resp. $S'$ (i.e. *locally constructable*), then the base change morphism is an isomorphism.

Why is that? Is that an easy fact?