Abstract Pathogen adaptation to multiple selective pressures challenges our ability to control their spread. Here we analyze the evolutionary dynamics of pathogens spreading in a heterogeneous host population where selection varies periodically in space. We study both the transient dynamics taking place at the front of the epidemic and the long-term evolution far behind the front. We identify five types of epidemic profiles arising for different levels of spatial heterogeneity and different costs of adaptation. In particular, we identify the conditions where a generalist pathogen carrying multiple adaptations can outrace a coalition of specialist pathogens. We also show that finite host populations promote the spread of generalist pathogens because demographic stochasticity enhances the extinction of locally maladapted pathogens. But higher mutation rates between genotypes can rescue the coalition of specialists and speed up the spread of epidemics for intermediate levels of spatial heterogeneity. Our work provides a comprehensive analysis of the interplay between migration, local selection, mutation, and genetic drift on the spread and on the evolution of pathogens in heterogeneous environments. This work extends our fundamental understanding of the outcome of the competition between two specialists and a generalist strategy (single- vs. multiadapted pathogens). These results have practical implications for the design of more durable control strategies against multiadapted pathogens in agriculture and in public health.