Is rational connectedness a constructible property?

Question

Let $f:X\rightarrow S$ be a morphism of varieties (say, over an algebraically closed field $k$). Is the locus of points above which the fibers of $f$ are rationally connected $\{s\in S:X_s \text{ is rationally connected}\}$ a constructible set in $S$? I'm wondering about this question because I read in an article a proof where the author seemed to have used the fact the the fiber of a morphism $g:V\rightarrow \mathbb{A}_k^n$ above every dimension $1$ point is rationally connected if we know that the generic fiber is rationally connected. I can see it can be deduced if being rational connected is a constructible property. If not, how do we transfer the rational connectedness to general fibers from the generic fiber as above?