This paper concentrates on investigating ergodicity and stability for generalised Markov branching processes with resurrection. Easy checking criteria including several clear-cut corollaries are established for ordinary and strong ergodicity of such processes. The equilibrium distribution is given in an elegant closed form for the ergodic case. The probabilistic interpretation of the results is clear and thus explained.

Previous work on the concept of a limiting conditional age distribution of a discrete-state continuous-time Markov process with one absorbing state is generalised. The generalisation allows this process to have a finite number of absorbing states and the associated return process to have an arbitrary initial distribution on the transient states of the absorbing process. If the return process is ρ-recurrent, possesses the strong ratio limit property and satisfies some further requirements then the limiting age distribution exists. The proof of this result requires a new representation of the ρ-invariant measure of the return process.

The following examples are treated, (a) finite state space birth-death processes, (b) Markov branching processes and the linear death process, and (c) the linear birth and death process with killing.