We consider the critical Galton–Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.

This result brings a new perspective on Vatutin’s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.

A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

A branching process is described which contains some better-known simple branching processes as special cases. Variables examined are the rate of development, the number of individuals currently alive, and the duration of the process. The apparent dynamics of collapsing supercritical realisations are isolated.

The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.

Characteristics of some collapsing supercritical stochastic population processes where an embedded Galton–Watson process can be isolated are shown to be identical to those of some associated subcritical process.