The gap isn't "something" and therefore cannot move or be created.
Current evidence shows that the universe expands faster than light. So spacetime moves faster than light but since it has no mass, it's not bound by the physical limits. It's hard to visualise because it's space itself that expands.
Similar to your example. If you put a beacon in a moving car, the light's speed won't be (speed of light+speed of car) because the speed of light is the limit.
Combining the 2 points above: It's the principle by which LIGO works. The gravitational ripples affect space itself but since anything that affects space also affects anything in it, you cannot observe the stretches and compressions of the waves. However, light's speed is constant and therefore will take longer/shorter time to reach the other side of the tube affected by the gravitational wave.