  Consider a population where individuals give birth at constant rate during
their lifetimes to i.i.d. copies of themselves. Individuals bear clonally
inherited types, but (neutral) mutations may happen at the birth events. The
smallest subtree containing the genealogy of all the extant individuals at a
fixed time \tau, is called the coalescent point process. We enrich this process
with the history of the mutations that appeared over time, and call it the
marked coalescent point process. With the help of limit theorems for L\'evy
processes with marked jumps established in a previous work (arXiv:1305.6245),
we prove the convergence of the marked coalescent point process with large
population size and two possible regimes for the mutations - one of them being
a classical rare mutation regime, towards a multivariate Poisson point process.
This Poisson point process can be described as the coalescent point process of
the limiting population at \tau, with mutations arising as inhomogeneous
regenerative sets along the lineages. Its intensity measure is further
characterized thanks to the excursion theory for spectrally positive L\'evy
processes. In the rare mutations asymptotic, mutations arise as the image of a
Poisson process by the ladder height process of a L\'evy process with infinite
variation, and in the particular case of the critical branching process with
exponential lifetimes, the limiting object is the Poisson point process of the
depths of excursions of the Brownian motion, with Poissonian mutations on the
lineages.
