Let $X$ be a fixed curve (e.g. a Noetherian, projective scheme of dimension 1, of finite type over an algebraically closed field $k$) and let $S$ be an arbitrary parameter scheme over $k$. Let $D \subset X \times S$ be a flat family over $S$ of subschemes of $X$, of relative dimension 0 and degree $d$, with ideal sheaf $\mathcal I$.
Consider now the clolsed subscheme $D' \subset X \times S$ defined by the ideal $\mathcal I^n$. In my mind, $D' = nD$ in some geometric sense. Is $D'$ again flat over $S$?