We discuss the natural Lagrangian and Eulerian formulations of multi-agent deterministic optimal control problems, analyzing their relations with a novel Kantorovich formulation. We exhibit some equivalence results among the various representations and compare the respective value functions, by combining techniques and ideas from optimal transport, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. (In collaboration with Giulia Cavagnari, Stefano Lisini and Carlo Orrieri)[-]