  In this note we continue the study of nonlocal interaction dynamics on a
sequence of infinite graphs, extending the results of [Esposito et. al 2023+]
to an arbitrary number of species. Our analysis relies on the observation that
the graph dynamics form a gradient flow with respect to a non-symmetric
Finslerian gradient structure. Keeping the nonlocal interaction energy fixed,
while localising the graph structure, we are able to prove evolutionary
{\Gamma}-convergence to an Otto-Wassertein-type gradient flow with a
tensor-weighted, yet symmetric, inner product. As a byproduct this implies the
existence of solutions to the multi-species non-local (cross-)interacation
system on the tensor-weighted Euclidean space
