In this talk, the evolution of a system of two species with nonlinear mobility on a graph with nonlocal interactions is discussed. We provide a rigorous interpretation of the interaction system as a gradient flow in a Finslerian setting. This not only extends the recent results of Esposito et. al. (2021) to systems of interacting species, but also translates the theory of gradient flows with concave, nonlinear mobilities to this setting. Weakening the notion of Minkowski norm and nonlocal gradient, in the spirit of Agueh (2011), the geometric interpretations and the analysis are carried over to p-Wasserstein-like distances.