Recent advances in materials processing technologies highlight the need to understand solidification kinetics in multicomponent alloys. Using a finite interface dissipation phase field model, we investigate planar front solidification rates in ternary alloys, used as a model system, as a function of various mass transport processes and nonequilibrium conditions. Simulation results reveal that the off-diagonal diffusion coefficients play an important role in controlling the solid–liquid (S–L) interface velocity and solute partition coefficients. Specifically, under various rapid solidification conditions negative values for the off-diagonal diffusion coefficients increase the solute partition coefficients due to the depletion of liquid phase concentrations ahead of the S–L interface. Given an undercooling, the S–L interface velocity increases with decreasing the off-diagonal diffusion coefficients. In broad terms, our work quantifies the role of coupled mass transport processes and nonequilibrium effects in solidification rates of multicomponent alloys.