A discrete agent-based model on a periodic lattice of arbitrary dimension is considered. Agents move to nearest-neighbor sites by a motility mechanism accounting for general interactions, which may include volume exclusion. The partial differential equation describing the average occupancy of the agent population is derived systematically. A diffusion equation arises for all types of interactions and is nonlinear except for the simplest interactions. In addition, multiple species of interacting subpopulations give rise to an advection-diffusion equation for each subpopulation. This work extends and generalizes previous specific results, providing a construction method for determining the transport coefficients in terms of a single conditional transition probability, which depends on the occupancy of sites in an influence region. These coefficients characterize the diffusion of agents in a crowded environment in biological and physical processes.