  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.
