Specialization map between stratified bundles and pro-\'etale fundamental group

Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of the pro-\'etale fundamental group of the closed fibre and the category of stratified bundles on the geometric generic fibre. By Tannakian duality, this functor induces a morphism between the corresponding affine group schemes. We show that this morphism is a lifting of the specialization map, constructed by Grothendieck, between the \'etale fundamental groups.