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

Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. 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.