Your statement is true, but it is important to emphasise that they travel at the speed of light in vacuum, this is important because the universe is only almost

Your statement is true, but it is important to emphasise that they travel at the speed of light in vacuum, this is important because the universe is only almost empty and electro-magnetic radiation actually travels slower than the speed of light in vacuum in this medium. This means that one would expect for light to arrive a few seconds after the detection of the gravitational wave. The statement is also only true for perturbations with respect to a fixed background metric. In principle it is possible for space-time to expand much faster than the speed of light (this is believed to have happened just after the big-bang).