  The well-posedness of a multi-population dynamical system with an entropy
regularization and its convergence to a suitable mean-field approximation are
proved, under a general set of assumptions. Under further assumptions on the
evolution of the labels, the case of different time scales between the agents'
locations and labels dynamics is considered. The limit system couples a
mean-field-type evolution in the space of positions and an instantaneous
optimization of the payoff functional in the space of labels.
