Mobility Metrics and Data Sources in Apple Health

I have an iPhone XS updated to the latest iOS and an Apple Watch Series 5 updated to the latest WatchOS. When looking at the data sources in the mobility metrics in the Health App on the iPhone, the watch is recording stairs up, and stairs down but the iPhone is recording walking asymmetry, step length, double support time, walking speed and six-minute walk. Shouldn't the watch be recording this? I've tried adding the watch as another data source in the iPhone measurements but there is no option for this.