Anarchist Memes

If the "fall of Mongolia" refers to the Yuan Dynasty, then the Han very obviously ruled China both prior and after that.

It depends on how they are interpreting Mongol. There are a couple ways to interpret Mongol depending on how exacting you wish to be. The most specific is just the Mongol empire, the period prior to the establishment of the yuan dynasty. The mongol dynasty which includes the yuan dynasty and the rest of the kaganates. Or the most general, the modern vernacular for tribal steppe people.

If they are just talking about ethnic groups originating in Manchuria then they are correct. The Qin dynasty was the first dynasty of imperial China, and it originated from the Manchu people prior to the Han Dynasty. These are ethnically the same people who would eventually establish the Yuan and Qing dynasties .

I don't think you understand exactly how long the beef between Manchu and Han goes back, or the modern and contemporary attempts by the Han to obscure their ethnic contributions via historical revisionism.

Modern Han chauvinism has been recognized as a problem for the leaders of China since the Taiping rebellion where you start to hear quotes like "China is the China of the Chinese. We compatriots should identify ourselves with the China of the Han Chinese."