  We introduce Wasserstein-like dynamical transport distances between
vector-valued densities on the real line. The mobility function from the scalar
theory is replaced by a mobility matrix, that is subject to positivity and
concavity conditions. Our primary motivation is to cast certain systems of
nonlinear parabolic evolution equations into the variational framework of
gradient flows. In the first part of the paper, we investigate the structural
properties of the new class of distances like geodesic completeness. The second
part is devoted to the identification of $\lambda$-geodesically convex
functionals and their $\lambda$-contractive gradient flows. One of our results
is a generalized McCann condition for geodesic convexity of the internal
energy. In the third part, the existence of weak solutions to a certain class
of degenerate diffusion systems with drift is shown. Even if the underlying
energy function is not geodesically convex w.r.t. our new distance, the
construction of a weak solution is still possible using de Giorgi's minimizing
movement scheme.
