The Stacks project

Proof. For every scheme $T$ over $B$ the morphism $\pi _ T : X_ T \to Y_ T$ is a closed immersion. Then $\pi _{T, *}$ is an equivalence of categories between $\mathit{QCoh}(\mathcal{O}_{X_ T})$ and the full subcategory of $\mathit{QCoh}(\mathcal{O}_{Y_ T})$ whose objects are those quasi-coherent modules annihilated by the ideal sheaf of $X_ T$, see Morphisms of Spaces, Lemma 67.14.1. Since a qotient of $(\pi _*\mathcal{F})_ T$ is annihilated by this ideal we obtain the bijectivity of the map $\mathrm{Quot}_{\mathcal{F}/X/B}(T) \to \mathrm{Quot}_{\pi _*\mathcal{F}/Y/B}(T)$ for all $T$ as desired. $\square$