We consider a class of Crump-Mode-Jagers processes with interaction, constructed by removing a newly born offspring with a probability that depends on the age structure of the population at its birth time. We prove a law of large numbers for the tree structure of the process in a local topology, and show how this result condenses several other limit theorems (convergence of the empirical age distribution, of ancestral lineages). Beyond this specific example, our work illustrates a more general principle that we formalise. As in standard propagation of chaos, the trees generated by typical individuals become independent as the number of individuals goes to infinity. This allows us to express the distribution of the local tree structure around a typical individual in terms of a time-inhomogeneous branching process, which we call a non-linear branching process.