  We consider a system of $n$ nonlocal interaction evolution equations on
$\mathbb{R}^d$ with a differentiable matrix-valued interaction potential $W$.
Under suitable conditions on convexity, symmetry and growth of $W$, we prove
$\lambda$-geodesic convexity for some $\lambda\in\mathbb{R}$ of the associated
interaction energy with respect to a weighted compound distance of Wasserstein
type. In particular, this implies existence and uniqueness of solutions to the
evolution system. In one spatial dimension, we further analyse the qualitative
properties of this solution in the non-uniformly convex case. We obtain, if the
interaction potential is sufficiently convex far away from the origin, that the
support of the solution is uniformly bounded. Under a suitable Lipschitz
condition for the potential, we can exclude finite-time blow-up and give a
partial characterization of the long-time behaviour.
