  We consider a Moran model with two allelic types, mutation and selection. In
this work, we study the behaviour of the proportion of fit individuals when the
size of the population tends to infinity, without any rescaling of parameters
or time. We first prove that the latter converges, uniformly in compacts in
probability, to the solution of an ordinary differential equation, which is
explicitly solved. Next, we study the stability properties of its equilibrium
points. Moreover, we show that the fluctuations of the proportion of fit
individuals, after a proper normalization, satisfy a uniform central limit
theorem in $[0,\infty)$. As a consequence, we deduce the convergence of the
corresponding stationary distributions.
