  We study the convergence of an $N$-particle Markovian controlled system to
the solution of a family of stochastic McKean-Vlasov control problems, either
with a finite horizon or Schr\"odinger type cost functional. Specifically,
under suitable assumptions, we prove the convergence of the value functions,
the fixed-time probability distributions, and the relative entropy of their
path-space probability laws. These proofs are based on a Benamou-Brenier type
reformulation of the problem and a superposition principle, both of which are
tools from the theory of optimal transport.
