We report a theoretical overview of the magnetic domain wall behavior under an electric current in infinitely-long nanotubes with azimuthal magnetization, combining the $1$D analytic model and micromagnetic simulations. We highlight effects that, besides spin-transfer torques already largely understood in flat strips, arise specifically in the tubular geometry: the \OErsted field and curvature-induced magnetic anisotropy resulting both from exchange and material growth. Depending on both the geometry of the tube and the strength of the azimuthal anisotropy, Bloch or N\'eel walls arise at rest, resulting in two regimes of motion largely dominated by either spin-transfer torques or the \OErsted field. We determine the Walker breakdown current in all cases, and highlight the most suitable parameters to achieve high domain wall speed.