As the Circuits are located in the soul, a fusion of body and mind is required during the transplant.
One of those methods is through the establishment of a mystic pass between two magi from which the Circuits can transferred. Like any other organ, the number of Magic Circuits that one possesses is determined at birth and cannot decrease or increase naturally, meaning that those from older lineages will be more powerful, but it is possible to change that number artificially through a number of methods. Their Quality and Quantity are ranked through letters like Magecraft, and their Composition often varies between magi. In these cases, destruction of the brain is necessary to stop the Circuits. There are rare occasions in which the Circuits continue working even after the practitioner’s body has ceased all vital functions. Because organic activity is necessary for the operation of Circuits, it is a common mistake to think that they are literally a part of the magus body. Magic Circuits do not change and once damaged they cannot be repaired. They are composed of core components, which are the actual Circuits so to say, and bypasses that connect those components to the brain.
Magic Circuits reside within the Magus’s Soul, and what is found on the body is just a physical expression of them that stretches itself throughout the body in a fashion similar to the nervous system. Various states of Shirou's Magic Circuit.