Let $f: X \to S$ be a proper morphism (not necessarily smooth!, but perhaps flat) and $S$ be an excellent, regular, affine, integral variety over a finite field (I do not know which of these properties are really necessary).
Does $\{s \in S : X_s := f^{-1}(s) \text{ is (geometrically) integral}\}$ contain an open subset [edit: this was: "non-empty open subset"]?
(The same question for $\{s \in S : X_s := f^{-1}(s) \text{ is (geometrically) connected}\}$ seems to have a positive answer by Stein factorisation.)