Phase III of Sewall Wright's shifting-balance process involves the spread of a superior genotype throughout a structured population. However, a number of authors have suggested that this sort of adaptive change is unlikely under biologically plausible conditions. We studied relevant mathematical models, and the results suggest that the concerns about phase III of the shifting-balance process are justified, but only if environmental conditions are stable. If environmental conditions change in a way that alters species range, then phase III can be effective, leading to an enhancement of adaptedness throughout a structured population.