Stacks are inserted at a year old (or probably upon creation in regards to clones), so I assume there's some sort of basic medical procedure that goes along with it. But the meths are switching sleeves through needlecasting, basically projecting their consciousness to their new sleeve. I'm unsure of the process that would initiate this, but that's the premise of how Rei was able to animate her clones the way that she did.

I know this is an old post, but I was checking out this wiki after recently finishing the first season and thought I’d contribute. My understandings of the stacks is that they’re a device that houses the consciousness and records new information from the brain and body of the sleeve. A stack, like the previous comment explains, is placed into every person at 1 year old and every alternate sleeve (synth, clone, etc.) at its creation. The individual stacks are mostly used as part of a sleeve, as it seems that consciousness is transfered between stacks and their sleeves more often than individual stacks containing someone’s consciousness are surgically placed into another sleeve. When a consciousness is needlecast, a transfer of consciousness takes place where the consciousness is moved, not copied as that’s illegal, onto a new stack in another sleeve. We mostly only see stacks when a sleeve dies, but even then I think the consciousness is usually transfered from the stack directly into a new sleeve, rather than physically swapping stacks. Therefore, when it comes to clones, the consciousness is automatically moved from the last body into the next one (especially if it’s backed up like Rei’s).