We present a multilayer Saint-Venant system for the numerical simulation of free surface density-stratified flows over variable topography. The proposed model formally approxi- mates the hydrostatic Navier-Stokes equations with a density that varies depending on the spatial and temporal distribution of a transported quantity such as temperature or salinity. The derivation of the multilayer model is obtained by a Galerkin-type vertical dis- cretization of the Navier-Stokes system with piecewise constant basis functions. In con- trast with classical multilayer models in the literature that assume immiscible fluids, we allow here for mass exchange between layers. We show that the multilayer system admits a kinetic interpretation, and we use this result to formulate a robust finite volume scheme for its numerical approximation. Several numerical experiments are presented, including simulations of wind-driven stratified flows.