Depth of $F$-singularities and base change of relative canonical sheaves

For a characteristic $p > 0$ variety $X$ with controlled $F$-singularities, we state conditions which imply that a divisorial sheaf is Cohen-Macaulay or at least has depth $\geq 3$ at certain points. This mirrors results of Kollár for varieties in characteristic zero. As an application, we show that that relative canonical sheaves are compatible with arbitrary base change for certain families with sharply $F$-pure fibers.