Hold a US passport and want to fly into Beijing from and to S. Korea. Can we use visa free transit?

Will be staying in S. Korea on extended visit but want to travel to other Asian countries during our trip. We would like to use the free transit to travel with our daughter (also a US citizen) who is working in S Korea, to take a short China tour. Do we have to fly back to a country other than Korea? If not can the other country be a transit country with final destination in Korea?