  Using the Minimal Model Program, any degeneration of K-trivial varieties can
be arranged to be in a Kulikov type form, i.e. with trivial relative canonical
divisor and mild singularities. In the hyper-K\"ahler setting, we can then
deduce a finiteness statement for monodromy acting on $H^2$, once one knows
that one component of the central fiber is not uniruled. Independently of this,
using deep results from the geometry of hyper-K\"ahler manifolds, we prove that
a finite monodromy projective degeneration of hyper-K\"ahler manifolds has a
smooth filling (after base change and birational modifications). As a
consequence of these two results, we prove a generalization of Huybrechts'
theorem about birational versus deformation equivalence, allowing singular
central fibers. As an application, we give simple proofs for the deformation
type of certain geometric constructions of hyper-K\"ahler manifolds (e.g.
Debarre--Voisin or Laza--Sacc\`a--Voisin). In a slightly different direction,
we establish some basic properties (dimension and rational homology type) for
the dual complex of a Kulikov type degeneration of hyper-K\"ahler manifolds.
