Let $S$ be a scheme of finite type over the complex numbers $\mathbb{C}$ and let $X\subset S\times\mathbb{P}^r$ be a projective family over $S$.
In the book "Geometry of Algebraic Curves Volume II" by Arbarello, Cornalba and Griffiths, Chapter 1, Section 7 it is described how to construct the relative Hilbert scheme $\text{Hilb}_{X/S}^{p(t)}$ parametrizing pairs $(s,Y)$ of a points $s\in S$ and a subscheme $Y\subset X_s$ with Hilbert polynomial $p(t)$.
It is described as a closed subscheme of $S\times \text{Hilb}_r^{p(t)}$ so, in particular, it comes with a projection $\text{Hilb}_{X/S}^{p(t)}\to S$.
Now my question is, under which circumstances is $\text{Hilb}_{X/S}^{p(t)}$ flat over $S$? For example, is this true if $X$ is flat over $S$?
In the specific example I'm trying to work out, $S=\mathbb{P}^N$ is the projective space parametrizing degree $d$ curves in $\mathbb{P}^2$, $X\subset S\times\mathbb{P}^2$ is the universal family over $S$ and $p(t)\equiv n$ is constant, so the fiber of $\text{Hilb}_{X/S}^n$ over $[C]\in S$ is the Hilbert scheme of $n$ points in the curve $C$. Is this true in this case?