Frobenius split subvarieties pull back in almost all characteristics

Abstract: Let $X$ and $Y$ be schemes of finite type over $\mathrm{Spec}\ \mathbb{Z}$ and let $\alpha: Y \to X$ be a finite map. We show the following holds for all sufficiently large primes $p$: If $\phi$ and $\psi$ are any splittings on $X \times \mathrm{Spec}\ F_p$ and $Y \times \mathrm{Spec}\ F_p$, such that the restriction of $\alpha$ is compatible with $\phi$ and $\psi$, and $V$ is any compatibly split subvariety of $(X \times \mathrm{Spec}\ F_p, \phi)$, then the reduction $\alpha^{-1}(V)^{\mathrm{red}}$ is a compatibly split subvariety of $(Y \times \mathrm{Spec}\ F_p, \psi)$. This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.