The "ringdown" that was detected after the merger of two black holes by LIGO in 2015 has been compared to the damping of a rung bell.
Why doesn't the merged black hole ring perpetually? What force dampens the ringing?
Gravitational waves carry energy, momentum, and angular momentum away to infinity, thereby damping the ringing and allowing the merged hole to settle down into a Kerr metric. The amount of energy radiated can be enormous... a significant fraction of the mass-energy of the merging black holes. For the first LIGO event, three solar masses was lost as gravitational radiation!
The ringdown can be studied using perturbation theory around a Kerr metric. The analysis does not typically involve “forces” because there is no matter to experience a force. However, one can think of the radiation as causing some kind of “recoil” causing the spacetime of the hole to settle down.