Establishing a versatile, fast and reliable magnetic memory technology is a giant bottleneck for cryogenic computing since present-day room-temperature solutions either cease to work or consume too much power. The long-term goal of superconducting spintronics has been to overcome this bottleneck by creating magnetic memories with equal-spin triplet supercurrent driven through them to control their magnetization direction. This path has been hampered by the short spin relaxation length and strong anisotropy in ferromagnets. Here we show how the supercurrent driven generation of spin accumulation in a superconductor/magnetic insulator bilayer together with Gilbert damping of magnetization lead to a motion of magnetic domain walls. This manifests as a local voltage across the wall, which allows its position to be identified. Associated with this voltage and the current, there is Joule power which is dissipated via the Gilbert damping. The power required to maintain domain wall motion is orders of magnitude smaller than in the normal state, where most of the power is wasted in maintaining the current.