  We study the large time behavior of Fujita-Kato type solutions to the
Vlasov-Navier-Stokes system set on $\mathbb{T}^3 \times \mathbb{R}^3$. Under
the assumption that the initial so-called modulated energy is small enough, we
prove that the distribution function converges to a Dirac mass in velocity,
with exponential rate. The proof is based on the fine structure of the system
and on a bootstrap analysis allowing to get global bounds on moments.
