The Kingman coalescent is an important and well-studied process in population genetics modelling the ancestry of a sample of individuals. In this talk weak convergence results are presented that characterise asymptotic properties of the Kingman coalescent under parent dependent mutations, as the sample size grows to infinity. It is shown that the sampling probability satisfies a power-law and we derive the asymptotic behaviour of transition probabilities of the block counting jump chain. For the normalised jump chain and number of mutations between types a limiting process is derived consisting of a deterministic component, describing the limit of the block counting jump chain, and independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. Finally, the results are extended to characterise the asymptotic performance of popular importance sampling algorithms, such as the Griffiths-Tavare algorithm and the Stephens-Donnelly algorithm. This is joint work with Martina Favero.