We propose and analyze a natural extension of the Moreau sweeping process: given a family of moving convex sets (C(t))t, we look for the evolution of a probability density ρt, constrained to be supported on C(t). We describe in detail three cases: in the first, particles do not interact with each other and stay at rest unless pushed by the moving boundary; in the second they interact via a maximal density constraint ρ ≤ 1, so that they are not only pushed by the boundary, but also by the other particles; in the third case particles are submitted to Brownian diffusion, reflected along the moving boundary. We prove existence, uniqueness and approximation results by using techniques from optimal transport, and we provide numerical illustrations.