I will present two stability results concerning the asymptotic behavior of solutions to classical collisionless models arising in kinetic theory. First, I will present an asymptotic stability result for the exterior of the Schwarzschild family of black holes as a solution to the Einstein-massless Vlasov system, assuming spherical symmetry. We exploit the normal hyperbolicity of the trapped set to obtain decay in time of the energy momentum tensor. As a toy model, I will also speak about an asymptotic stability result for small data solutions to the Vlasov-Poisson system with an unstable trapping potential (joint work with Anibal Velozo).