Effective motives with and without transfers in characteristic p

We prove the equivalence between the category RigDMéteff(K,Q) of effective motives of rigid analytic varieties over a perfect complete non-archimedean field K   and the category RigDAFrobéteff(K,Q) which is obtained by localizing the category of motives without transfers RigDAéteff(K,Q) over purely inseparable maps. In particular, we obtain an equivalence between RigDMéteff(K,Q) and RigDAéteff(K,Q) in the characteristic 0 case and an equivalence between DMéteff(K,Q) and DAFrobéteff(K,Q) of motives of algebraic varieties over a perfect field K. We also show a relative and a stable version of the main statement.