  We consider contractivity for diffusion semigroups w.r.t. Kantorovich ($L^1$
Wasserstein) distances based on appropriately chosen concave functions. These
distances are inbetween total variation and usual Wasserstein distances. It is
shown that by appropriate explicit choices of the underlying distance,
contractivity with rates of close to optimal order can be obtained in several
fundamental classes of examples where contractivity w.r.t. standard Wasserstein
distances fails. Applications include overdamped Langevin diffusions with
locally non-convex potentials, products of these processes, and systems of
weakly interacting diffusions, both of mean-field and nearest neighbour type.
