All the princes get a dream education and there's no reason to waste it, so everyone but the heir finds another noble to work under by the time the throne changes hands. They stop using the title of prince and their living situation would be completely up to their employer. It's also not unheard of for them to instead open small schools for commoners. Their children aren't really considered nobles but can still be invited by the Imperial Council to be compared to the next batch of princes if the princes don't seem terribly impressive.