Diffusion and rheology in a mixture of two simple or one simple and one complex fluids are studied systematically in the framework of GENERIC modeling. The main result is a set of intrinsically consistent equations governing the time evolution of the overall mass density, overall momentum density (this equation involves explicit formulae for the scalar hydrodynamic pressure and the extra stress tensor), relative concentration of one fluid, diffusion flux, and the microstructure of the complex fluid. In the process of solving these equations we recover well-known results concerning the influence of stresses on diffusion and obtain new results concerning the influence of diffusion on stresses.