Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}_X, \mathcal{M}_Y$.

Does one have for free an induced pullback morphism $f^*: \mathcal{M}_X \to \mathcal{M}_Y$?

I guess the question reduces to: if $S$ is a base scheme and $E$ is a family of sheaves flat over $S$ on $X$, and if $f_S: Y_S \to X_S$ is the induced morphism, then is $f_S^* E$ still a flat family of coherent sheaves? (as usual, for ugly $S$ coherent $E$ should be replaced by quasi-coherent (locally) of finite presentation).

Thanks