Counterexample: Let $X$ be the plane and $Y$ the blow-up of $X$ at the origin. Consider the tautological family of length 1 skyscraper sheaves on $X$, parametrized by $X$. This means $S=X$, and $E=\Delta_\ast\mathscr{O}_X$, where $\Delta:X\to X\times X$ is the diagonal. Then $f_S^\ast E=(\Gamma_f)_\ast\mathcal{O}_Y$, where $\Gamma_f:Y\to Y\times X$ is the graph of $f$. This is not flat over $X$.
This makes sense, geometrically: there is no way to pull back the skyscraper sheaf at the origin in $X$ to a skyscraper sheaf in $Y$, in a way which is compatible with families of skyscraper sheaves, for example along lines through the origin.