Convergence of weighted branching processes

We study the long-term behavior of weighted multi-type branching processes, focusing on extending classical laws of large numbers and martingale convergence to settings with infinitely many weighted particles, arbitrary type spaces and non-geometric rescaling. We demonstrate applications to Galton-Watson trees indexed by random weights and by random kernels, convergence in Wasserstein distance of the underlying mean semi-group, and convergence of ergodic averages along lineages.