Let $f:X\rightarrow S$ be a finite type morphism of noetherian schemes and let $S$ be integral. Let $F$ be a coherent sheaf on $X$.
Let $T$ be an integral closed subscheme of $S$. Then we can apply generic flatness to $f^*F$ on $X_T\rightarrow T$. Then there is an open dense set $V\subset T$ such that $f^*F|_V$ is flat over $V$.
Can we find an open set $U$ of $S$ such that $F|_U$ is flat over $U$ and $V\subset U$?
I had another question about flat families that I am adding to this. If $W$ is the maximal subset of $S$ such that $F|_W$ is $W$-flat. Then is $W$ open in general? I think not. But what if $f$ is a faithfully flat morphism?