I will present a ‘good’ compactification of the relative Hilbert scheme of points associated to the smooth locus of a (simple) degeneration of varieties. This compactification is obtained by GIT-methods, and yields a new, refined, approach to earlier constructions of J. Li and B. Wu. I will also discuss how these results can be used to study degenerations of irreducible holomorphic symplectic varieties. This is joint work with M. Gulbrandsen and K. Hulek.