  We generalize the classical semi-continuity theorem for GIT (semi)stable loci
under variations of linearizations to a relative situation of an equivariant
projective morphism from X to an affine base S. As an application to moduli
problems, we consider degenerations of Hilbert schemes, and give a conceptual
interpretation of the (semi)stable loci of the degeneration families
constructed by Gulbrandsen-Halle-Hulek.
