Existence of log canonical closures

Let $f:X\to U$ be a projective morphism of normal varieties and $(X,\mathrm{\Delta })$ a dlt pair. We prove that if there is an open set $U^0\subset U$, such that $(X,\mathrm{\Delta }){\times }_U{U}^0$ has a good minimal model over $U^0$ and the images of all the non-klt centers intersect $U^0$, then $(X,\mathrm{\Delta })$ has a good minimal model over $U$. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.