  We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability
in terms of one parameter subgroups of a linearly reductive group G over a
field k, to the relative situation of an equivariant, projective morphism X ->
Spec A to a noetherian k-algebra A. We also extend the classical projectivity
result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is
projective. As an example of applications to moduli problems, we consider
degenerations of Hilbert schemes of points.
