Motion-damping systems and methods that include motion-damping systems are disclosed herein. The motion-damping systems include a flexible body that is configured to extend within a gap that is defined between a base structure and an attached component. The motion-damping systems further include a magnetic assembly and a ferromagnetic body. One of the magnetic assembly and the ferromagnetic body is located within the flexible body and the other of the magnetic assembly and the ferromagnetic body is operatively affixed to a selected one of the base structure and the attached component. The magnetic assembly and the ferromagnetic body are oriented such that a magnetic force therebetween retains the flexible body in physical contact with the selected one of the base structure and the attached component. The methods include methods of installing and/or operating the motion-dampening systems.