The Stacks project

Proof. By Morphisms, Lemma 29.8.5, dominant morphisms are preserved under pullback by open morphisms, hence by flat morphisms locally of finite presentation (Morphisms, Lemma 29.25.10). The other direction is easier. Indeed, being dominant is clearly Zariski local on the base. Next, let $S' \to S$ be a surjective morphism of schemes (not necessarily flat or locally of finite presentation), and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is dominant. Then $X' \to S'\to S$ is a composition of two dominant morphisms, hence dominant. Since this is also the composition $X' \to X \to S$, it follows that $X\to S$ is dominant. $\square$