  We study the evolution of a system of two species with nonlinear mobility and
nonlocal interactions on a graph whose vertices are given by an arbitrary,
positive measure. To this end, we extend a recently introduced
$2$-Wasserstein-type quasi-metric on generalized graphs, which is based on an
upwind-interpolation, to the case of two-species systems, concave, nonlinear
mobilities, and $p\ne 2$. We provide a rigorous interpretation of the
interaction system as a gradient flow in the Finslerian setting, arising from
the new quasi-metric.
