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A micromagnetic study is carried out to analyze the influence of the spin-orbit Rashba field on the current-driven domain wall dynamics along a ferromagnetic strip with high perpendicular magnetocrystalline anisotropy, taking into account both surface roughness and thermal effects. Similarly to the creep and flow regimes seen in recent experiments, the domain wall velocity increases exponentially and linearly for low and high currents, respectively. The induced transverse Rashba field stabilizes the Bloch structure, extending the rigid mobility regime toward high currents and allowing higher velocities than in single-layer or symmetric stacks with negligible Rashba field.