The adult (parent/guardian) responsible for the unaccompanied minor on the departure side must stay at the departure gate until the plane is in the air so in a delay the UM would stay with that adult until the UM boards the plane. UMs are only allowed on flights without a plane change so the child would stay on the plane in the event of a stop on the way to the destination. While it's possible to have a mechanical problem that would require a plane change at the stopover, it's certainly not common and I believe Southwest personnel would escort the minor to the other plane. That's not explicitly stated in the UM policy online. At the destination, the UM is escorted off the plane and a designated adult (parent/guardian) meets the minor at the gate. Southwest checks ID and hands over the UM at the gate.
More details and other info can be found here: https://www.southwest.com/html/customer-service/family/unaccompanied-minors-pol.html
