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The dynamics of a magnetic domain wall (DW) under a transverse magnetic field Hy are investigated in two-dimensional (2D) Co/Ni microstrips, where an interfacial Dzyaloshinskii-Moriya interaction (DMI) exists with DMI vector D lying in +y direction. The DW velocity exhibits asymmetric behavior for ±Hy; that is, the DW velocity becomes faster when Hy is applied antiparallel to D. The key experimental results are reproduced in a 2D micromagnetic simulation, which reveals that the interfacial DMI suppresses the periodic change of the average DW angle φ even above the Walker breakdown and that Hy changes φ, resulting in a velocity asymmetry. This suggests that the 2D DW motion, despite its microscopic complexity, simply depends on the average angle of the DW and thus can be described using a one-dimensional soliton model. These findings provide insight into the magnetic DW dynamics in 2D systems, which are important for emerging spin-orbitronic applications.