In the spirit of Duquesne and Winkel (2007) and Berestycki et al. (2011), we show that supercritical continuous-state branching process with a general branching mechanism and general immigration mechanism is equivalent in law to a continuous-time Galton-Watson process with immigration (with Poissonian dressing). The result also helps to characterise the limiting backbone decomposition which is predictable from the work on consistent growth of Galton-Watson trees with immigration in Cao and Winkel (2010).
Kyprianou, A. E., & Ren, Y. X. (2012). Backbone decomposition for continuous-state branching processes with immigration. Statistics & Probability Letters, 82(1), 139-144. https://doi.org/10.1016/j.spl.2011.09.013