Gravitational friction is just a slowing down due to gravitational radiation. Say, when you walk, you radiate about 10^-57 watt of gravitational waves. Jupiter radiation while orbiting Sun is about 450 watt. It is small in small grav field and large near event horizon.

If you do not want to solve GR equations, you may get some reasonable solution out of Newtonian equation including dr/dt friction term in it.

You get orbit precessions and fancy spiraling orbits near black holes (but not in the VERY strong fields where you have to use GR).