Elucidation of homoepitaxial growth mechanisms on vicinal non-polar surfaces of GaN is highly important for gaining an understanding of and control thin film surface morphology and properties. Using first-principles calculations, we study the step-flow growth in m-plane GaN based on atomic row nucleation and kink propagation kinetics. Ga-N dimer adsorption onto the m-plane is energetically more favorable than that of Ga and N isolated adatoms. Therefore, we have treated the dimers as the dominant growth species attached to the step edges. By calculating the free energies of sequentially attached Ga-N dimers, we have elucidated that the a-step edge kink growth proceeds by parallel attachment rather than by across the step edge approach. We found a series of favorable configurations of kink propagation and calculated the free energy and nucleation barriers for kink evolution on five types of step edges (a, +c, -c, +a + c, and -a - c). By changing the chemical potential μGa and the excess chemical potential Δμ, the growth velocities at the five types of edges are controlled by the corresponding kink pair nucleation barrier E* in their free energy profiles. To explore the kink-flow growth instability observed at different Ga/N flux ratios, calculations of kink pairs on the incompact -c and +c-step edges are further performed to study their formation energies. Variations of these step edge morphologies with a tuned chemical environment are consistent with previous experimental observations, including stable diagonal ±a ± c-direction steps. Our work provides a first-principles approach to explore step growth and surface morphology of the vicinal m-plane GaN, which is applicable to analyze and control the step-flow growth of other binary thin films.