  Consider a branching Markov process with values in some general type space.
Conditional on survival up to generation $N$, the genealogy of the extant
population defines a random marked metric measure space, where individuals are
marked by their type and pairwise distances are measured by the time to the
most recent common ancestor. In the present manuscript, we devise a general
method of moments to prove convergence of such genealogies in the Gromov-weak
topology when $N \to \infty$.
  Informally, the moment of order $k$ of the population is obtained by
observing the genealogy of $k$ individuals chosen uniformly at random after
size-biasing the population at time $N$ by its $k$-th factorial moment. We show
that the sampled genealogy can be expressed in terms of a $k$-spine
decomposition of the original branching process, and that convergence reduces
to the convergence of the underlying $k$-spines.
  As an illustration of our framework, we analyse the large-time behavior of a
branching approximation of the biparental Wright-Fisher model with
recombination. The model exhibits some interesting mathematical features. It
starts in a supercritical state but is naturally driven to criticality. We show
that the limiting behavior exhibits both critical and supercritical
characteristics.
