Apart from extra attacks gained from BAB, there's no order in which your attacks must be taken during a Full Attack action.
Therefore, if I'm a high-level Monk with three natural weapons (and using your interpretation), on my Full Attack I can:
1. Attack with US, and be Flurrying
2. Attack with a claw, and not be Flurrying
3. Attack with US strike, and be Flurrying
4. Attack with a claw, and not be Flurrying
5. Attack with a kama, and be Flurrying
6. Attack with a bite, and not be Flurrying
I can switch between "Flurrying right now" and "Not Flurrying right now" multiple times within the same Full Attack action.