This is "Ghost in the Shell"stuff.
Can the Abstractions have their OWN abstractions? Would those be a copy of the ORIGINAL's consciousness, or an entirely NEW consciousness... Or something else,
In GitS the show several known kinds of this happening. Most obvious is the Major who extensively uses multiple memory backup cyberbrains until presumably the last piece of "meat" dies. The Tachikomas continuously extract and analyze data and modify their own THINKING as they go. They were able to inhabit a remote server and switch bodies at will. Presumably their code contained the keys to doing it again.
Then you had other events like Ryesomes on the network, servers that held a particular point of view even as users came and went.. Kind of like Slashdot. They also had "stand alone complex" in which the larger group of "people" created the same idea multiple times without people collaborating. In that respect, keeping memories and personality "alive" on the network could be a way for humans collective experience to continue even when the idea carrier dies.. To keep the "spark" of those ideas available for the future human race.