The electrokinetic mixing, as a powerful technique in microfluidic devices, is widely used in many applications. In this study, a more general dynamic model, which consists of Poisson equation, Nernst-Planck equation and Navier-Stokes equations, is used to describe the electrokinetic mixing of non-Newtonian fluids in microchannels. Furthermore, a coupled multiple-relaxation-time (MRT) lattice Boltzmann (LB) framework is developed to solve this complicated multi-physics transport phenomenon. In numerical simulations, we mainly consider the effects of the arrangement of nonuniform surface potentials and the power-law index on the mixing efficiency and the volumetric flow rate. Numerical results show that the mixing efficiency and the volumetric flow rate of shear-thinning fluids are higher than that of shear-thickening fluids under the same condition. It is also shown that for both types of fluids, there should be a balance between the mixing and liquid transport in electrokinetic microfluidics.