Evolution is a fundamentally population level process in which variation, drift and selection produce both temporal and spatial patterns of change. Statistical model fitting is now commonly used to estimate which kind of evolutionary process best explains patterns of change through time using models like Brownian motion, stabilizing selection (Ornstein–Uhlenbeck) and directional selection on traits measured from stratigraphic sequences or on phylogenetic trees. But these models assume that the traits possessed by a species are homogeneous. Spatial processes such as dispersal, gene flow and geographical range changes can produce patterns of trait evolution that do not fit the expectations of standard models, even when evolution at the local‐population level is governed by drift or a typical OU model of selection. The basic properties of population level processes (variation, drift, selection and population size) are reviewed and the relationship between their spatial and temporal dynamics is discussed. Typical evolutionary models used in palaeontology incorporate the temporal component of these dynamics, but not the spatial. Range expansions and contractions introduce rate variability into drift processes, range expansion under a drift model can drive directional change in trait evolution, and spatial selection gradients can create spatial variation in traits that can produce long‐term directional trends and punctuation events depending on the balance between selection strength, gene flow, extirpation probability and model of speciation. Using computational modelling that spatial processes can create evolutionary outcomes that depart from basic population‐level notions from these standard macroevolutionary models.