Basically surgery. The process is sorta like organ transfers. That's even a comparison made a few times. Obviously the process is more, well, magical than a medical surgery since it involves both the body and the soul, but every time it is mentioned the idea given is that it is a slow, involved, complicated process. Like a surgery.

@LorenerdSupreme

Yes, however, it was mentioned that the Crest can be transferred from a dead person to a living one (Rin and Tokiomi). In addition, it takes several attempts before it's fully transferred, which is what I don't understand.