On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules (See Stacks Project). I want to consider the following generalization.
Let $f:X\to S$ be a (projective flat) morphism of Noetherian schemes. Let $\mathcal F$ be a quasi-coherent $\mathcal O_X$-module. Assume $\mathcal F$ is flat over $S$. Then $\mathcal F$ is the filtered colimit of its coherent $\mathcal O_X$-submodules which are flat over $S$.
Indeed I am interested in the following conjecture, which naturally generalizes to the above.
Let $X$ be a projective scheme over a field $\Bbbk$. Let $A$ be an Artinian local $\Bbbk$-algebra with residue field $\Bbbk$. Denote $X_A:=X\times_\Bbbk\operatorname{Spec}(A)$. Let $\mathcal F$ be quasi-coherent on $X_A$, flat over $A$. Then $\mathcal F$ is the filtered colimit of its coherent submodules that are flat over $A$.
Are there any references?