Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be said for the "universal closed subscheme" $Z\subset \mathbb{P}^n_S\times_S \operatorname{Hilb}^p$? It is easy to see that $Z$ represents a modified Hilbert functor which further specifies a section, so it seems possible to me that a careful reading of Hartshorne's proof would establish the connectedness of $Z$ as well.
Even if $Z$ is not connected, it might be useful to know the answer to the following: if $k$ is a field, does every $k$-point of $Z$ which corresponds to an abelian variety (with the marked section) lie in the same connected component? Theorem 6.14 from GIT seems relevant here, but Mumford already assumes the base to be connected. I've copied this theorem below for convenience.
GIT Theorem 6.14: Let $S$ be a connected, locally noetherian scheme. Let $\pi:X\rightarrow S$ be a smooth projective morphism, and let $\varepsilon:S\rightarrow X$ be a section of $\pi$. Assume that for one geometric point $s$ of $S$, the fiber $X_s$ of $\pi$ is an abelian variety with identity $\varepsilon(s)$. Then $X$ is an abelian scheme over $S$ with identity $\varepsilon$.
