Neighbors are going to become more aggressive until the colony can prove itself independently capable of self-defense. In some cases this means the colony is slowly swamped until it dies, others that is stabilizes, and in a rare few the Empire's prohibitions on expansion might be removed and the colony aggressively expands into areas forbidden to it previously.

Politically there is going to be tension between traditionalists wanting to retain old institutions and relationships, and innovators looking to improve the system or adapt to circumstances. This often takes a long time, centuries or in some places millennia and they then give increasingly symbolic nods to the way things used to be. Maybe they still pretend the Empire is in charge after the Empire itself dies, or claim to be the Emperor of a 20 mile stretch and act like they will eventually take back the center.