  In this paper we consider the transport--Stokes system, which describes the
sedimentation of particles in a viscous fluid in inertialess regime. We show
existence of Lagrangian solutions to the Cauchy problem with $L^1$ initial
data. We prove uniqueness of solutions as a corollary of a stability estimate
with respect to the 1-Wasserstein distance for solutions with initial data in a
Yudovich-type refinement of $L^3$, with finite first moment. Moreover, we
describe the evolution starting from axisymmetric initial data. Our approach is
purely Lagrangian.
