The talk will be devoted to the analysis of a variational problem that describes the optimal evolution of a probability measure, subject to density constraints and nonlocal potential of Riesz type. The problem arises in the study of some first order Mean Field type control problems, with density constraints and aggregating interactions. Under suitable symmetry assumptions, I will discuss the existence of periodic orbits, and their convergence to hetero clinics, which connect different ground states of the stationary problem in an infinite time horizon. Finally, I will discuss some qualitative properties of the discrete counterpart of the problem, which involves the optimal evolution of a finite number of particles that are subject to distance constraints.