Native in US. We mow around the best clumps until they start looking ratty, then just mow with the rest of the grass. A good portion of our grass is Tradescantia. Not really welcome in a bed though, as said, it's seedy/spready. If you feel industrious, you can line them up along an edge somewhere, or packed into a fantastic clump, to mow later than the rest of the lawn, semi-cultivated, so to speak.
