In this article we propose a type-dependent branching model with mutation and competition, which can be used to model phylodynamic patterns of a virus population. For any two virus particles, the competition kernel depends on the particles’ types and the total mass of the population. We introduce our individual-based model as a measure-valued process and discuss possible scaling regimes. We then model the evolving phylogenies as stochastic processes with values in the space of marked metric measure spaces. For that we rely on the genetic information given through the number of nucleotide substitutions separating the virus particles. Finally, we construct a two- level branching model that describes branching with competition within splitting cells. In all cases the large population limit of these models solves a martingale problem. Uniqueness of solutions is verified for the first and third model. The techniques for showing uniqueness include novel formulations of Girsanov’s theorem in the presence of jumps as well as a novel duality relation in the two-level-model. In the second model uniqueness is open but conjectured to hold under restricted conditions on the branching and competition rates. For this model the proof of existence of the limit relies on a new method of showing the compact containment condition for stochastic processes with values in metric measure spaces that are a priori not ultra-metric.