Organisms interact with each other mostly over local scales, so the local density experienced by an individual is of greater importance than the mean density in a population. This simple observation poses a tremendous challenge to theoretical ecology, and because nonlinear stochastic and spatial models cannot be solved exactly, much effort has been spent in seeking effective approximations. Several authors have observed that spatial population systems behave like deterministic nonspatial systems if dispersal averages the dynamics over a sufficiently large scale. We exploit this fact to develop an exact series expansion, which allows one to derive approximations of stochastic individual-based models without resorting to heuristic assumptions. Our approach makes it possible to calculate the corrections to mean-field models in the limit where the interaction range is large, and it provides insight into the performance of moment closure methods. With this approach, we demonstrate how the buildup of spatiotemporal correlations slows down the spread of an invasion, prolongs time lags associated with extinction debt, and leads to locally oscillating but globally stable coexistence of a host and a parasite.