  Motivated by conjectures of Demailly, Green-Griffiths, Lang, and Vojta, we
show that several notions related to hyperbolicity behave similarly in
families. We apply our results to show the persistence of arithmetic
hyperbolicity along field extensions for projective normal surfaces with
nonzero irregularity. These results rely on the mild boundedness of
semi-abelian varieties. We also introduce and study the notion of
pseudo-algebraic hyperbolicity which extends Demailly's notion of algebraic
hyperbolicity for projective schemes.
