The fabric of spacetime itself sets the ultimate speed limit. Nothing can locally move through it faster than light. For example, gravitational waves ripple across the universe at light speed.
Anything that exists within spacetime is bound by this rule. The only odd exception people point to is quantum entanglement, but while the correlations appear instantaneous, they can’t be used to send information faster than light. Sending matter is distant second.
So, if we ever hope to travel faster than light, we wouldn’t do it by "outrunning" gravity. Instead, we’d need to find a way to manipulate spacetime itself, like bending, warping, or reshaping it ... since that, in the first place itself, is what is defining the limits of motion.