An asymmetric displacement of domain walls during their oscillatory motion and an asynchronous change in the velocities of neighboring domain walls have been revealed in iron garnet crystals with different characters of magnetic anisotropy in an alternating harmonic magnetic field at field amplitudes lower than the field initiating the drift motion of the domain structure. It has been established that in a bipolar pulsed magnetic field the displacement of domain walls in the direction of drift exceeds the displacement of domain walls in the direction opposite to drift. A theoretical model in which asymmetry in the oscillations of domain walls and drift appear due to anisotropy in the attenuation parameter is proposed.