In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for ``almost all'' initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we study the flow in the space of measures induced by the continuity equation: we prove existence, uniqueness and stability properties of the flow in this infinite-dimensional space, in the same spirit of the theory developed in the case when the state space is Euclidean.