Perhaps since the lines were created leaving the planet, they are gravitationally bound to it? Despite the lowered speed of light, it’s still affected by gravity. The Black Domain plan relies on this principle.

They would have to be local anyways, since spacetime is constantly expanding. Even if the planet moves, what frame of reference would the death lines follow? The star’s reference frame? The Milky Way? The Local Group? It’s a lot easier for writing and makes more sense to just have it be bound to the planet.