To first answer your question I think it is prudent to identify why SpaceX is moving the Center of Mass(CoM) in the first place.
Space capsules since the Mercury days have needed heat shields for reentry. The high speeds and relatively flat shapes cause compressive heating which leads to exceedingly high temperatures. While heat shield materials like AvCoat and PICA-X are marvelous materials that can withstand high temperature, extended exposure to reentry plasma can cause the structure to fail. Engineers identified that a ballistic reentry from Lunar Return velocities(11km/s) would create unmanageable heating effects. Their solution was to change the ballistic trajectory into a semi-controlled descent. The Apollo Command Module had a CoM offset from its axis of symmetry. That created an aerodynamically stable "edge" of the capsule that could be controlled by rotating the craft. This allowed the capsule to generate lift and slow the acceleration from reentry.
Apollo used its Reaction Control System(RCS) to change the orientation of its CoM.
Apollo EDL Plan