Magnetic domain walls can be moved by spin-polarized currents due to spin-transfer torques. This opens the possibility to use them in spintronic memory devices as, e.g., in racetrack storage. Naturally, in miniaturized devices domain walls can get very close to each other and affect each other's dynamics. In this work, two separated domain walls in different layers which interact via an interlayer exchange coupling have been considered. One of these walls is moved by a spin-polarized current. Depending on several parameters as the current density, the interlayer coupling or the pinning potential, the combined dynamics of the two domain walls can change very strongly, allowing for, e.g., a correlated motion of the walls. In addition, a more subtle effect appears as a suppression of the Walker breakdown accompanied by an increase of the domain wall velocity.