  We consider the controllability problem for the continuity equation where the
dynamics is governed by a driftless control-affine system. Under suitable
regularity conditions, the controllability of the system is a sufficient
condition for the controllability of the continuity equation by means of
time-varying feedback controls. Moreover, we show that there exist controls
such that the flow of the control system is the optimal transport map, for the
2-Wasserstein distance, between two given probability measures.
