I will present upcoming linear and non-linear stability results concerning the asymptotic behavior of collisionless many-particle systems on black hole exteriors. On the one hand, I will discuss the decay properties of massive Vlasov fields on Schwarzschild spacetime. On the other hand, I will discuss an asymptotic stability result for the exterior of Schwarzschild  spacetime as a solution to the Einstein-massless Vlasov system, assuming spherical symmetry. These works are motivated by the black hole stability problem with Vlasov matter.