  We show, for a class of discrete Fleming-Viot (or Moran) type particle
systems, that the convergence to the equilibrium is exponential for a suitable
Wassertein coupling distance. The approach provides an explicit quantitative
estimate on the rate of convergence. As a consequence, we show that the
conditioned process converges exponentially fast to a unique quasi-stationary
distribution. Moreover, by estimating the two-particle correlations, we prove
that the Fleming-Viot process converges, uniformly in time, to the conditioned
process with an explicit rate of convergence. We illustrate our results on the
examples of the complete graph and of N particles jumping on two points.
