A theorem on meromorphic descent and the specialization of the pro-\'etale fundamental group

Given a Noetherian formal scheme $\widehat{X}$ over $\mathrm{S}\mathrm{p}\mathrm{f}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\widehat{X}$. Using this we construct a specialization functor from the category of continuous representations of the pro-'etale fundamental group of the special fiber to the category of $F$-divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the 'etale fundamental groups. We also express the pro-'etale fundamental group of a connected scheme $X$ of finite type over a field as coproducts and quotients of the free group and the 'etale fundamental groups of the normalizations of the irreducible components of $X$ and those of its singular loci.