Let $X\to S$ be a proper morphism of Noetherian schemes. Let $\mathcal E,\mathcal F$ be coherent $\mathcal O_X$-modules, flat over $S$. Let $g:T\to S$ be a morphism of coherent schemes. There is a base change map $$g^*\operatorname{Ext}_X^i(\mathcal E,\mathcal F)\to \operatorname{Ext}_{X\times_ST}^i\big(g^*\mathcal E,g^*\mathcal F\big). $$
I now want to know whether there is any results on this base change map. For example:
If the base change map is surjective, then is it an isomorphism?
and
Is there any open subscheme $S'\subseteq S$ such that the base change map is an isomorphism when $T\to S$ factors through $S'$?