  We study the structure of Frobenius splittings (and generalizations thereof)
induced on compatible subvarieties $W \subseteq X$. In particular, if the
compatible splitting comes from a compatible splitting of a divisor on some
birational model $E \subseteq X' \to X$ (ie, this is a log canonical center),
then we show that the divisor corresponding to the splitting on $W$ is bounded
below by the divisorial part of the different as studied by Kawamata, Shokurov,
Ambro and others. We also show that difference between the divisor associated
to the splitting and the divisorial part of the different is largely governed
by the (non-)Frobenius splitting of fibers of $E \to W$. In doing this
analysis, we recover an $F$-canonical bundle formula by reinterpretting
techniques common in the theory of Frobenius splittings.
