  We study the large time behavior of small data solutions to the
Vlasov-Navier-Stokes system set on $\Omega \times \mathbb{R}^3$, for a smooth
bounded domain $\Omega$ of $\mathbb{R}^3$, with homogeneous Dirichlet boundary
condition for the fluid and absorption boundary condition for the kinetic
phase. We prove that the fluid velocity homogenizes to $0$ while the
distribution function concentrates towards a Dirac mass in velocity centered at
$0$, with an exponential rate. The proof, which follows the methods introduced
in [Han-Kwan - Moussa - Moyano, arXiv:1902.03864v2], requires a careful
analysis of the boundary effects. We also exhibit examples of classes of
initial data leading to a variety of asymptotic behaviors for the kinetic
density, from total absorption to no absorption at all.
