  We investigate the asymptotic in $N$ of the mixing times of a Markov dynamics
on $N-1$ ordered particles in an interval. This dynamics consists in resampling
at independent Poisson times each particle according to a probability measure
on the segment formed by its nearest neighbours. In the setting where the
resampling probability measures are symmetric, the asymptotic of the mixing
times were obtained and a cutoff phenomenon holds. In the present work, we
focus on an asymmetric version of the model and we establish a cutoff
phenomenon. An important part of our analysis consists in the derivation of a
hydrodynamic limit, which is given by a non-linear Hamilton-Jacobi equation
with degenerate boundary conditions.
