  The goal of this article is to study the limit of the empirical distribution
induced by a mutation-selection multi-allelic Moran model, whose dynamic is
given by a continuous-time irreducible Markov chain. The rate matrix driving
the mutation is assumed irreducible and the selection rates are assumed
uniformly bounded. The paper is divided into two parts. The first one deals
with processes with general selection rates. For this case we are able to prove
the propagation of chaos in $\mathbb{L}^p$ over the compacts, with speed of
convergence of order $1/\sqrt{N}$. Further on, we consider a specific type of
selection that we call additive selection. Essentially, we assume that the
selection rate can be decomposed as the sum of three terms: a term depending on
the allelic type of the parent (which can be understood as selection at death),
another term depending on the allelic type of the descendant (which can be
understood as selection at birth) and a third term which is symmetric. Under
this setting, our results include a uniform in time bound for the propagation
on chaos in $\mathbb{L}^p$ of order $1/\sqrt{N}$, and the proof of the
asymptotic normality with zero mean and explicit variance, for the
approximation error between the empirical distribution and its limit, when the
number of individuals tend towards infinity. Additionally, we explore the
interpretation of the Moran model with additive selection as a particle process
whose empirical distribution approximates a quasi-stationary distribution, in
the same spirit as the Fleming\,--\,Viot particle systems. We then address the
problem of minimising the asymptotic quadratic error, when the time and the
number of particles go to infinity.
