  The results in this paper provide new information on asymptotic properties of
classical models: the neutral Kingman coalescent under a general
finite-alleles, parent-dependent mutation mechanism, and its generalisation,
the ancestral selection graph. Several relevant quantities related to these
fundamental models are not explicitly known when mutations are parent
dependent. Examples include the probability that a sample taken from a
population has a certain type configuration, and the transition probabilities
of their block counting jump chains. In this paper, asymptotic results are
derived for these quantities, as the sample size goes to infinity. It is shown
that the sampling probabilities decay polynomially in the sample size with
multiplying constant depending on the stationary density of the Wright-Fisher
diffusion and that the transition probabilities converge to the limit of
frequencies of types in the sample.
