We study a broad class of nonlocal advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the nonlocal presence of others. Our model allows for different nonlocal interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of nonlocal advection-diffusion equations. We demonstrate that solutions can blow up in finite time when the detection radius becomes zero, i.e. when the system is local, thus showing that nonlocality is essential for the global existence of solutions. We verify our results with numerical simulations on 2D spatial domains.