The Stacks project

Proof. By Morphisms, Lemma 29.8.4, quasi-compact dominant morphisms are preserved by flat pullback. The other direction is easier. Indeed, quasi-compactness is fpqc local on the base by Lemma 35.23.1. Being dominant is clearly Zariski local on the base. Finally, let $S' \to S$ be a surjective morphism of schemes (not necessarily flat), 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 composite of two dominant morphisms, hence dominant. Since this is also the composite $X' \to X \to S$, it follows that $X\to S$ is dominant. $\square$