This paper proves existence of a global weak solution of the inhomogeneous (i.e., non-constant density) incompressible Navier-Stokes system with mass diffusion. The system is well-known as the Kazhikhov-Smagulov model. The major novelty of the paper is to deal with the Kazhikhov-Smagulov model possessing the nonconstant viscosity without any simplification of higher order nonlinearity. Any global weak solution is shown to have a long time behavior that is consistent with mixing phenomena of miscible fluids. The results also contain a new compactness method of the Aubin-Lions-Simon type.