We provide an extension of the theory of flows in Euclidean spaces associated to wekly differentiable velocity fields, replacing the Euclidean state space with the space of probability measure. This way, the continuity equation can be viewed as an ODE in the space of probability measures and we provide some well-posedness and stability results. We illustrate, as an example and a basic motivation for the development of the theory, the problem of convergence of Wigner transforms in Quantum Mechanics.