Cold black sky holds the secret first. A planet torn from its native orbit keeps moving with almost the same velocity it had before ejection, and that smooth transit through interstellar space actually protects the miniature system of satellites locked to it.
The counterintuitive part is simple. Star loss does not erase local gravity. Around any massive body sits a gravitational domain known as a Hill sphere, the region where its pull dominates over distant competitors. When a planet is flung outward by a close gravitational encounter, its moons already reside deep inside that sphere. Their orbital velocity, set by the planet’s mass and radius, exceeds the weak differential tug from whatever distant star remains in the background.
Radiation and tides are the real losers here. Stripped of stellar heating, the planet cools; yet thermal loss does not touch orbital energy on human or geological timescales. As the system coasts through largely empty space, there is almost nothing to sap angular momentum. Only a later close pass near another star or massive planet can truncate the Hill sphere enough to pry moons away, so most rogue worlds wander on as intact miniature solar systems, lights out but still bound.