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The motions of different types of domain walls (DWs) driven by direct current (dc) magnetic field in soft magnetic nanostripes were calculated. The DWs exhibit steady and transient motions in the low fields and oscillations of their internal structure above a critical field. A developed soliton model of the walls explains their dynamics by the motions of a limited number of magnetic topological solitons having nonzero gyrovectors such as vortex and antivortex. The model predicts the reduced DW velocity and critical field in the low-field regime, and increased wall-oscillation frequency in nanostripes, compared to the Walker solution for bulk magnets. The critical field and velocity are determined by the nanostripe thickness and width, whereas the oscillation frequency depends only on the field strength.