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In this article we prove exactness of the homotopy sequence of overconvergent fundamental groups for a smooth and projective morphism in characteristic p. We do so by first proving a corresponding result for rigid analytic varieties in characteristic 
$0$, following dos Santos [dS15] in the algebraic case. In characteristic p, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result. We then use this to deduce a Lefschetz hyperplane theorem for convergent fundamental groups, as well as a comparison theorem with the étale fundamental group.
