Frobenius split subvarieties pull back in almost all characteristics

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

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