  In this work, we study the mean field Schr\"odinger problem from a purely
probabilistic point of view by exploiting its connection to stochastic control
theory for McKean-Vlasov diffusions. Our main result shows that the mean field
Schr\"odinger problem arises as the limit of ``standard'' Schr\"odinger
problems over interacting particles. Due to the stochastic maximum principle
and a suitable penalization procedure, the result follows as a consequence of
novel (quantitative) propagation of chaos results for forward-backwards
particle systems. The approach described in the paper seems flexible enough to
address other questions in the theory. For instance, our stochastic control
technique further allows us to solve the mean field Schr\"odinger problem and
characterize its solution, the mean field Schr\"odinger bridge, by a
forward-backward planning equation.
