Measure-valued and coalescent processes have a long and rich history in the field of population genetics. Through duality, these processes are intimately linked and the study of one often leads to insights in the other. In this thesis we investigate the small-time behaviour of backwards-in-time genealogical processes to establish a diffusion limit at t = 0. Forwards in time we consider linear combinations of an i.i.d. collection of Fleming-Viot processes, describing LLN and CLT style limits. In order to establish convergence in the second chapter, we require tightness. The duality of these processes and results from our first chapter are essential in proving this and much of the second chapter.