Wild birds are one of the main natural reservoirs for avian influenza viruses, and their migratory behavior significantly influences the transmission of avian influenza. To better describe the migratory behavior of wild birds, a system of reaction–advection–diffusion equations is developed to characterize the interactions among wild birds, poultry, and humans. By the next-generation operator, the basic reproduction number of the model is formulated. Then the threshold dynamic of the model is explored by some techniques including the theory of uniform persistence, internally chain transitive sets, and so on. Subsequently, the sensitivity analysis of parameters associated with the basic reproduction number is implemented. According to the temporal and spatial overlapping relationship between wild blue-winged ducks and poultry in North America, the effect of this relationship on the characteristic of spatial-temporal distribution of the viruses is well studied. Additionally, the risk of virus transmission from wild birds to poultry and humans is evaluated. The main results highlight that the basic reproduction number is more significantly affected by the parameters related to wild birds. Interestingly, the model output regarding the spatial distribution of poultry infections is consistent with the actual findings. Moreover, the risk of virus spillover from wild birds into poultry and humans varies with wild bird behavior and has a more substantial impact on poultry. Throughout this study, the critical risk points in the transmission process are identified, providing a theoretical basis for the prevention and control of avian influenza.