  The well-posedness of a non-local advection-selection-mutation problem
deriving from adaptive dynamics models is shown for a wide family of initial
data. A particle method is then developed, in order to approximate the solution
of such problem by a regularised sum of weighted Dirac masses whose
characteristics solve a suitably defined ODE system. The convergence of the
particle method over any finite interval is shown and an explicit rate of
convergence is given. Furthermore, we investigate the asymptotic-preserving
properties of the method in large times, providing sufficient conditions for it
to hold true as well as examples and counter-examples. Finally, we illustrate
the method in two cases taken from the literature.
