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