As regards the situation in Τibet, I’ve said elsewhere that history is a slippery ally when forced into the service of contemporary political disputes.
Let’s set aside the Mongols for a moment. They ruled an empire that stretched from Korea to Kiev, so one could use the Khans to make all kinds of territorial claims. Yuan (1279-1368) rule was extremely short in duration anyway, and most of the territories outside China proper were beyond Ming (1368-1644) control for nearly 300 years.
The Qing Dynasty (1644-1912) is a good place to start as the Manchus did maintain garrisons on the Τibetan plateau while administering the region through local elites. The Qing rulers, great patrons of Lamamism, consolidated their rule by maintaining cultural and religious ties with Τibet beyond mere military occupation. They also–generally but not always–ruled with a light touch, allowing relative autonomy in religious and cultural matters, which suited the situation quite well. The Qing Dynasty was, after all, a large, multi-ethnic empire, and maintaining order and peace in outlying territories was the utmost concern.
The problem is that the PRC is a nation-state, and the demands a nation-state places on its people are different than those of an empire.