  We study the magnetic dynamics of magnetoelectric antiferromagnetic thin
films, where an unconventional boundary ferromagnetism coexists with the bulk
N\'{e}el phase below the N\'{e}el temperature. The spin exchange between the
two order parameters yields an effective low-energy theory that is formally
equivalent to that of a ferrimagnet. Dynamics of domain walls and skyrmions are
analyzed within the collective variable approach, from which we conclude that
they behave as massive particles moving in a viscous medium subjected to a
gyrotropic force. We find that the film thickness can be used as a control
parameter for the motion of these solitons. In this regard, it is shown that an
external magnetic field can drive the dynamics of domain walls, whose terminal
velocity is tunable with the sample thickness. Furthermore, the classification
of the skyrmion dynamics is sensitive to the spatial modulation of the sample
thickness, which can be easily engineered with the present (thin-film)
deposition techniques. Current-driven spin transfer can trigger drifting orbits
of skyrmions, which can be utilized as racetracks for these magnetic textures.
