Summary: A modification of the branching stochastic process with immigration and with continuous states introduced by S. R. Adke and V. G. Gadag [in: C.C. Heyde (ed.), Branching processes. Proceedings of the first world congress 1993 in Varna, Bulgaria. New York: Springer-Verlag. Lect. Notes Stat. 99, 90–105 (1995; Zbl 0839.60077)] is considered. Limit theorems for the non-critical processes with or without non-stationary immigration and finite variance are proved. The subcritical case is illustrated with examples.