... at least in the case where the target is a rationally connected variety.

This question is a follow-up to question Constructing embedded families of curves with general moduli and Jason Starr's suggestion in the comments there.

Let $X$ be a smooth projective variety of dimension $\geq 3$ that is separably rationally connected. Fix $g \geq 2$, $r \geq 0$ and sufficiently positive curve class $\beta$. Consider the morphism $$ \mathcal{F} := (ev,\Phi): \mathcal{M}_{g,r}(X,\beta) \to X^r \times \mathcal{M}_{g,r} $$

What can be said about the topological properties of the fiber of $\mathcal{F}$? (for a general reference to topological properties of fibers of morphism, see : Structure of fundamental groups arising from smooth projective morphisms)

That is,

- What conditions do I need to impose to say something about the homotopy type of the general fiber? In particular, what restrictions can I impose to deduce that it is connected and simply connected outside a complex codimension 1 subset of $\mathcal{M}_{g,r} \times X^r$? The motivation for this question is to understand if we can lift classes in $\pi_1^{orb}(\mathcal{M}_{g,r})$ to $\pi_1(\mathcal{M}_{g,r}(X,\beta))$ using the standard fibration long exact sequence argument.
- I know the study of the rational connectedness of the fibers (which I guess is kind of an algebraic-geometry analogue of connectedness) of these map plays some kind of a rule in the study of rational connectedness of the target variety. Can somehow provide a reference for a good paper that summerizes what is currently known?
- Is there a known paper that studies the lift of the orbifold fundemental group of $\mathcal{M}_{g,r}$ in this context? That is, a lift of the MCG to the fundamental group of the Kontsevich moduli space of stable maps?