If a Roman citizen were to be imprisoned or otherwise removed from society (execution/banishment?) what would happen to their slaves?
Similarly, how would any other property be administered in their absence?
Well, the Romans didn't have prisons per se. They did have a few structures such as the Tullianum that we might call prisons today, but they were not meant for long-term incarceration. Instead, they were places to put criminals or captives awaiting execution. Alternative sentences aside from prison time for wealthy Romans included house arrest (often at their country villas), exile, or execution. The lower classes often faced forced labor, certain death in gladiatorial games, or death.
But what about the slaves of Romans sent into exile? One of the exile sentences was aquae et ignis interdictio (interdiction of water and fire), which required the confiscation of all goods by the state, presumably including slaves. However, many exiles such as Milo were attended by a group of slaves just as they were in Rome.