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　mu(Ga)　and　the　excess　chemical　potential　Delta　mu,　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.