We have modelled the surface diffusion and growth of BaO and SrO both in the homoepitaxial and heteroepitaxial (BaO on SrO and SrO on BaO) cases. The diffusion proceeds most favourably by an exchange mechanism involving the surface layer. When impurities are adsorbed on the surface this can lead to intermixing between the layers. This strongly suggests that ionic materials may not be grown on a substrate with a similar structure without significant intermixing. Island growth begins with the formation of individual clusters which grow and merge together.