Growth of Galton-Watson trees: immigration and lifetimes

We study certain consistent families $(F_\lambda)_{\lambda\ge 0}$ of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in $F_\lambda$. Specifically, consistency here refers to the property that for each $\mu\le\lambda$, the forest $F_\mu$ has the same distribution as the subforest of $F_\lambda$ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of $F_\lambda$ is coloured in black independently with probability $\mu/\lambda$. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov's (non-Markovian) generalisation of continuous-state branching renewal processes, and similar processes with immigration.