We study a new class of distances between Radon measures similar to those
studied in a recent paper of Dolbeault-Nazaret-Savar\'e [DNS]. These distances
(more correctly pseudo-distances because can assume the value $+\infty$) are
defined generalizing the dynamical formulation of the Wasserstein distance by
means of a concave mobility function. We are mainly interested in the physical
interesting case (not considered in [DNS]) of a concave mobility function
defined in a bounded interval. We state the basic properties of the space of
measures endowed with this pseudo-distance. Finally, we study in detail two
cases: the set of measures defined in $R^d$ with finite moments and the set of
measures defined in a bounded convex set. In the two cases we give sufficient
conditions for the convergence of sequences with respect to the distance and we
prove a property of boundedness.