  We investigate the evolution of barycenters of masses transported by
stochastic flows. The state spaces under consideration are smooth affine
manifolds with certain convexity structure. Under suitable conditions on the
flow and on the initial measure, the barycenter {Z_t} is shown to be a
semimartingale and is described by a stochastic differential equation. For the
hyperbolic space the barycenter of two independent Brownian particles is a
martingale and its conditional law converges to that of a Brownian motion on
the limiting geodesic. On the other hand for a large family of discrete
measures on suitable Cartan-Hadamard manifolds, the barycenter of the measure
carried by an unstable Brownian flow converges to the Busemann barycenter of
the limiting measure.
