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Spin-orbit torque (SOT) driven domain wall motion has attracted significant attention as the basis for a variety of spintronic devices due to its potential use as a high speed, low power means to manipulate the magnetic state of an object. While most previous attention has focused on ultrathin films wherein the material thickness is significantly less than the magnetic exchange length, recent reports have suggested unique dynamics may be achieved in intermediate and high thickness films. We used micromagnetic modelling to explore the role of the vertically non-uniform spin textures associated with the domain wall in nanowires of varying thickness on SOT driven domain wall motion. We found large velocity asymmetries between Bloch chiralities near the current density required for reversal of the Bloch component of the magnetization and linked these asymmetries to a gradual reorientation of the domain wall structure which drives a non-negligible, chiral Néel component of the domain wall. We further explored the influence of saturation magnetization, film thickness, the Dzyaloshinskii-Moriya interaction, and in-plane fields on domain wall dynamics. These results provide a framework for the development of SOT based devices based on domain wall motion in nanowires beyond the ultrathin film limit.