  Nonlocal cross-diffusion systems on the torus, arising in population dynamics
and neural networks, are analyzed. The global existence of weak solutions, the
weak-strong uniqueness, and the localization limit are proved. The kernels are
assumed to be positive definite and in detailed balance. The proofs are based
on entropy estimates coming from Shannon-type and Rao-type entropies, while the
weak-strong uniqueness result follows from the relative entropy method. The
existence and uniqueness theorems hold for nondifferentiable kernels. The
associated local cross-diffusion system is also discussed.
