andrew
It didn't "start out" as foreign. They always were- and they conquered Northern Song, not "China". Then they had their entire ruling class exterminated, brutally, by the Mongols. It's ironic, lots of nationalists preaching "Altaic brotherhood" but the most vicious conflicts of East Asia are generally between "Altaics" and their so-called brothers- like Khitan Liao/Jurchen Jin vs. Mongols and each other, Korea vs. Japan (x2), and it was only the so-called "Altaic" dynasties of China that invaded Korea.
Thormodr
If the Russians and Indians then called themselves Chinese, devoted themselves entirely to "Chinese" interests, became heavily Sinicized linguistically and culturally, yes. Otherwise no. In that same vein are the late Ottomans and Mughals "Turks" (meaning actual Turks, not people who live in Turkey) and "Mongols" respectively? No.
No, China was never "conquered" by any one single power- Chinese rebels permitted Mongols and Manchus to displace a previous dynasty, and they were sinicized as a result. There is only nationalism on the part of anti-Han revisionists who have a racial obsession.
China is not the only country conquered by Mongols which eventually assimilated them. The Mongols did the same thing in the Middle East, Northern India and much of Central Asia. The fact that the Mongol conquerors converted to Islam eventually doesn't mean that they never conquered much of the Middle East.