  We show the validity of two special cases of the four-dimensional Minimal
Model Program in characteristic $p>5$: for contractions to
$\mathbb{Q}$-factorial fourfolds and in families over curves ("semi-stable
mmp"). We also provide their mixed characteristic analogues. As a corollary, we
show that liftability of positive characteristic threefolds is stable under the
minimal model program, and that liftability of three-dimensional Calabi-Yau
varieties is a birational invariant. Our results are partially contingent upon
the existence of log resolutions.
