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.
"Critical Branching Processes with Nonhomogeneous Migration." Ann. Probab. 13 (3) 923 - 933, August, 1985. https://doi.org/10.1214/aop/1176992914