By faithfully flat descent for morphisms of schemes, if $f \colon X \to X_i$ is faithfully flat and $g\colon X\to Z$ is an arbitrary morphism of schemes, then there exists a unique morphism $h \colon Z_i \to Z $$h \colon X_i \to Z $ such that $g = h\cdot f $ if (and in fact only if) the pullbacks of $h$$g$ to $X \times_{X_i} X$ along the two projections $X \times_{X_i} X \to X$ are equal.
Your conditions "surjective with connected fibers" do not imply faithfully flat, but the projection $f$ is faithfully flat: Any morphism from a nonempty scheme to the spectrum of a field is faithfully flat so in particular $\prod_{j \neq i }X_j \to \operatorname{Spec} k$ is, and faithfully flat morphisms are preserved by base change, so $f$ is as well.
Your condition "constant on the fibers of $f$" implies that the pullbacks of $h$$g$ to $X \times_{X_i} X$ along the two projections $X \times_{X_i} X \to X$ are equal as long as $X \times_{X_i} X$ is reduced (for example if the $X_i$ are all reduced over a field of characteristic zero): In this case, it suffices to check the equality at every point of $X \times_{X_i} X$, which lies in the product of one of the fibers of $f$ with itself, so it suffices to check that the value of $h$$g$ at the two points in the fiber are the same, which is the same as constancy.
Being a little careful I think one can check the implication just using the assumption that $X_i$ is reduced. Your desired statement is certainly not true if $X_i$ is not reduced.
