Firstly I don't agree with a single word of the idea of "Bringing Mongolia back into the Chinese nation".
That is truly unreasonable and unrealistic.
Secondly, someone said that China was a part of the Mongol Empire.
However the fact is that there was not such country officially called "China".
The Mongolian conquered ancient Chinese nations such as Jing, Song, Western Xia and the others, but there wasn't a country was called "China".
You don't say that Prussia conquered Germany nor Mugal Empire conquered India. Same stories.
"China" 中國, a nation founded by Chinese people (Tibetan, Mongolian, Han, Hun, Manchurian etc, has become an officially nation's name since the Republic was established. China was never a part of Outer Mongolia, but Outer Mongolia was part of China until the Soviet toke over and the Khalkha people betrayed the Golden Family and the Central Government.
Besides there are 6 million Mongol ethnic in China nowadays (half of the Mongolians in the world), yet there are only 2.8 million population in Outer Mongolia.
These 2.8 million Outer Mongolian are not even the posterity of Genghis Khan, yet Inner Mongolian (Golden Family) are the direct posterity of Genghis Khan.
If China cannot claim the Yuan Dynasty, then nobody can, especially for Outer Mongolia.
China is a country of [Chinese people], not only for [Han Chinese].
Han Chinese is just a part of the Chinese family, besides, the concept of Han people has been changing for thousands years. The meaning of Han people nowadays is not even the same as the ancient meaning.