Stochastic models of closed populations predict eventual extinction with certainty. Consequently, their behavior is often characterized by the quasi-stationary state, i.e. the long-term distribution of population sizes conditional on non-extinction. In contrast, models which allow for immigration exhibit a regular stationary state. At the limit of a low immigration rate, a population is expected to alternate between three states: the quasi-stationary state of a closed population, the extinction state, and the transient phase during which a newly arrived immigrant either establishes a new population or fails to do so. We develop this argument into a simple and intuitive framework that can be used to assess the effect of immigration in a general class of population models. We exemplify the framework for models in which immigrants arrive either singly or in groups, for models with an Allee effect, for models with environmental stochasticity, and for models leading to metapopulation dynamics.