  We consider an approximating sequence of interacting population models with
branching, mutation and competition. Each individual is characterized by its
trait and the traits of its ancestors. Birth- and death-events happen at
exponential times. Traits are hereditarily transmitted unless mutation occurs.
The present model is an extension of the model used in [M\'el\'eard and Tran,
EJP, 2012], where for large populations with small individual biomasses and
under additional assumptions, the diffusive limit is shown to converge to a
nonlinear historical superprocess. The main goal of the present article is to
verify a compact containment condition in the more general setup of Polish
trait-spaces and general mutation kernels that allow for a dependence on the
parent's trait. As a by-product, a result on the paths of individuals is
obtained. An application to evolving genealogies on marked metric measure
spaces is mentioned where genealogical distance, counted in terms of the number
of births without mutation, can be regarded as a trait. Because of the use of
exponential times in the modeling of birth- and death-events the analysis of
the modulus of continuity of the trait-history of a particle plays a major role
in obtaining appropriate bounds.
