Transport of majority carriers across the potential barriers frequently found at grain boundaries in semiconductors is treated by considering sequential drift diffusion and thermionic emission processes. The regimes where the thermionic or drift-diffusion approximations can reasonably be made are readily determined from the analysis. Over a wide doping range, we find boundary transport to be under mixed control. Numerical calculations are made for a given barrier height with a range of bulk doping levels. A deconvolution scheme to determine defect-state densities is given and tested for a variety of densities of states.