Gravity, according to our current understanding in the framework of general relativity, does not have a specific range like some other fundamental forces. Unlike electromagnetic or nuclear forces, which diminish with distance, gravity theoretically extends infinitely across space. However, the strength of gravity does weaken as you move farther away from its source.
In the scenario you mentioned, where galaxies are moving away from us due to the expansion of space, their recession speed can indeed exceed the speed of light. This phenomenon is known as cosmological recession and is a consequence of the expansion of the universe. However, it is important to note that the recession speed of galaxies is not due to their physical motion through space but rather the expansion of space itself.
Despite the galaxies receding faster than the speed of light, their gravity does not become absolutely zero to us. Gravity is not limited by the speed of light; it can still have an effect across large distances. Even though we might not be able to observe or interact with those galaxies directly due to their extreme distances and recession speeds, their gravitational influence can still be felt. Gravity propagates at the speed of light, and the gravitational field generated by distant objects takes time to reach us. So, in principle, we would still be influenced by the gravitational effects of those distant galaxies, even if we cannot observe them directly.