We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.