If S is a noetherian scheme and π : Z → X a morphism of S-schemes, where X is proper over S and Z is quasi-projective over S, then the set-valued contravariant functor $\Pi_{Z/X/S}$ on locally noetherian S-schemes, which associates to any T the set of all sections of $π_T : Z_T → X_T$, is representable by an open subscheme of $Hilb_{Z/S}$. This is an exercise in Nitsure, "Construction of Hilbert and Quot Schemes", and the proof is similar to the construction of the scheme of morphisms "Mor". My questions are:

- What is known about this scheme? It should be locally noetherian and quasiprojective, right? Can we say anything more?
- Can this be generalized somewhat? For example, can we dispense with the "X proper" assumption?
- Is there an analogous statement in the analytic setting? By wich I mean, assuming π : Z → X a morphism of complex varieties, where X is compact and Z quasi-projective, for instance.

References would be nice, even to FGA if that is the right place to look.