  The Kingman coalescent is a fundamental process in population genetics
modelling the ancestry of a sample of individuals backwards in time. In this
paper, in a large-sample-size regime, we study asymptotic properties of the
coalescent under neutrality and a general finite-alleles mutation scheme, i.e.
including both parent independent and parent dependent mutation. In particular,
we consider a sequence of Markov chains that is related to the coalescent and
consists of block-counting and mutation-counting components. We show that these
components, suitably scaled, converge weakly to deterministic components and
Poisson processes with varying intensities, respectively. Along the way, we
develop a novel approach to generalise the convergence result from the parent
independent to the parent dependent mutation setting. This approach is based on
a change of measure and provides a new alternative way to address problems in
the parent dependent mutation setting, in which several crucial quantities are
not known explicitly.
