Most work in evolutionary game theory analyzes a deterministic adjustment process on a continuum of agents. However, both the assumption of a continuum and that of no randomness are approximations, so it is important to study the behavior of adjustment processes on a large but finite population subject to small but persistent stochastic shocks. This paper characterizes the properties of the invariant distribution of birth-death processes in the double limit as the population becomes infinitely large and the perturbation vanishingly small. We show that the order of these limits does not change the conclusions for processes with “strong basins,” which is the case when the unperturbed process is deterministic. In contrast, the order of limits does matter for processes with “weak basins,” where the unperturbed process is stochastic except at a finite number of points.