We experimentally study the structure and dynamics of magnetic domains in synthetic antiferromagnets based on Co/Ru/Co films. Dramatic effects arise from the interaction among the topological defects comprising the dual domain walls in these structures. Under applied magnetic fields, the dual domain walls propagate following the dynamics of bi-meronic (bi-vortex/bi-antivortex) topological defects built in the walls. Application of an external field triggers a rich dynamical response: The propagation depends on mutual orientation and chirality of bi-vortices and bi-antivortices in the domain walls. For certain configurations, we observe sudden jumps of composite domain walls in increasing field, which are associated with the decay of composite skyrmions. These features allow for the enhanced control of domain-wall motion in synthetic antiferromagnets with the potential of employing them as information carriers in future logic and storage devices.