This paper deals with a modification of Galton-Watson processes allowing random migration in the following way: with a probability $p_n$(in the nth generation) one particle is eliminated and does not take part in further evolution, or with a probability $r_n$ takes place immigration of new particles according to a p.g.f. $G(s)$, and, finally, with a probability $q_n$ there is not any migration, $p_n + q_n + r_n = 1, n = 0, 1, 2, \cdots$. We investigate a critical case when the offspring mean is equal to one and $r_nG'(1) \equiv p_n \rightarrow 0$. Depending on the rate of this convergence we obtain different types of limit theorems.
N. M. Yanev. K. V. Mitov. "Critical Branching Processes with Nonhomogeneous Migration." Ann. Probab. 13 (3) 923 - 933, August, 1985. https://doi.org/10.1214/aop/1176992914