  In this paper we define a relative rigid fundamental group, which associates
to a section $p$ of a smooth and proper morphism $f:X\rightarrow S$ in
characteristic $p$, a Hopf algebra in the ind-category of overconvergent
$F$-isocrystals on $S$. We prove a base change property, which says that the
fibres of this object are the Hopf algebras of the rigid fundamental groups of
the fibres of $f$. We explain how to use this theory to define period maps as
Kim does for varieties over number fields, and show in certain cases that the
targets of these maps can be interpreted as varieties.
