  In this paper, we consider a mean field game model inspired by crowd motion
in which several interacting populations evolving in $\mathbb R^d$ aim at
reaching given target sets in minimal time. The movement of each agent is
described by a control system depending on their position, the distribution of
other agents in the same population, and the distribution of agents on other
populations. Thus, interactions between agents occur through their dynamics. We
consider in this paper the existence of Lagrangian equilibria to this mean
field game, their asymptotic behavior, and their characterization as solutions
of a mean field game system, under few regularity assumptions on agents'
dynamics. In particular, the mean field game system is established without
relying on semiconcavity properties of the value function.
