Genealogical structure of multiplying media in bounded domains

Consider a model of radiation transport which can be represented as a branching Markov process in a bounded domain. Suppose we know the position and velocity of k particles as they exit the domain, what is the law of the genealogical tree that connects them? In particular, what can we say about the law of their most recent common ancestor. This will be the best estimate of the radiation source if we don’t know it.