In this study, the migration of microparticles towards the inertial equilibrium positions in a straight microchannel with a square cross section in the presence of an inhomogeneous oscillating electric field was examined. The dynamics of microparticles were simulated using the immersed boundary-lattice Boltzmann method of fluid-structure interaction simulation. Moreover, the lattice Boltzmann Poisson solver was applied to calculate the electric field required for calculation of the dielectrophoretic force using the equivalent dipole moment approximation. These numerical methods were implemented on a single GPU coupled with the AA pattern of storing distribution functions in memory to speed up the computationally demanding simulation of microparticles dynamics. In the absence of an electric field, spherical polystyrene microparticles migrate to four symmetric stable equilibrium positions corresponding to the sidewalls of the square cross-sectional microchannel. The equilibrium distance from the sidewall was increased by increasing the particle size. The equilibrium positions near electrodes disappeared and particles migrated to the other equilibrium positions far from the electrodes by the application of the high-frequency oscillatory electric field at voltages beyond a threshold value. Finally, a two-step dielectrophoresis-assisted inertial microfluidics methodology was introduced for particle separation based on the crossover frequencies and the observed threshold voltages of different particles. The proposed method exploited the synergistic effect of dielectrophoresis and inertial microfluidics methods to remove their limitations, allowing the separation of a broad range of polydisperse particle mixtures with a single device in a short time.