The Stacks project

Proof. By Properties of Stacks, Lemma 100.5.4 and Lemmas 101.25.4 and 101.27.11 it suffices to prove to the lemma after replacing $\pi$ by a base change with a surjective, flat, locally finitely presented morphism $\mathcal{Y}' \to \mathcal{Y}$. By Lemma 101.28.7 we may assume $\mathcal{Y} = U$ is an algebraic space and $\mathcal{X} = [U/G]$ over $U$. Then $U \to [U/G]$ is surjective, flat, and locally of finite presentation, see Lemma 101.28.6. This implies that $\pi$ is surjective, flat, and locally of finite presentation by Properties of Stacks, Lemma 100.5.5 and Lemmas 101.25.5 and 101.27.12. $\square$