In a generalized age-dependent branching process, immigrants are allowed to arrive according to a non-homogeneous Poisson process. A limit theorem is given for the critical case.
"A Problem Concerning Generalized Age-Dependent Branching Processes with Immigration." Ann. Math. Statist. 42 (3) 1121 - 1123, June, 1971. https://doi.org/10.1214/aoms/1177693344