While the Outer Space Treaty has some things to say about it (the Moon Treaty was never ratified, or even signed by many of the players), historically the rules of precedence for establishing claim over new lands has been:
1. First to spot it.
2. First to plant a flag on it (which historically implied setting foot)
3. First to set up a base or fort on it
4. First to establish a settlement (ie, permanent habitation) on it.
With "right of ownership" proceeding in the above order. Robotic flag planting as we've had since the mid 1960's might be step 1.5, which is where China is at. USA was at 3 for a brief time in 1969-72 (since the later Apollo missions had surface stays of several days) although disclaimed it with the "we came in peace for all mankind" verbiage on the landing plaques.
If/when China establishes a manned base on the Moon, is there going to be anyone in a position to argue about it (beyond stern words at the UN and threats to remove "Most Favored Nation" trading status) if they claim ownership?