  We introduce a new transportation distance between probability measures that
is built from a L\'evy jump kernel. It is defined via a non-local variant of
the Benamou-Brenier formula. We study geometric and topological properties of
this distance, in particular we prove existence of geodesics. For translation
invariant jump kernels we identify the semigroup generated by the associated
non-local operator as the gradient flow of the relative entropy w.r.t. the new
distance and show that the entropy is convex along geodesics.
