Do we need a visa for mulitple trips while in Shanghai?

We are flying to SH (some with UK passports and some with US passports) and plan to stay one night before we take a day trip to Beijing returning back to SH the following day. we will then spend another night in SH before our 8 night cruise to Japan which will return to SH where we will spend one more night and head back to the US (via Dubai) the next day. To clarify, we will technically be in China (SH & Beijing) for 3 days before we leave for our cruise to Japan. Are we eligible for visa-free transit?