Genghis Khan understood muslims. That is why the only cities he utterly destroyed and razed to the ground and killed all the citys inhabitants were Islamic cities. With all other non-muslim cities, he captured them and added them to his empire while allowing the citizens to keep their religions and cultures.
Wikipedia (linking to other sources there) says this about Genghis Khan:
He was famous for being merciless. Known for killing boys and men of captured cities; and kidnapping the woman and girls. To a question about the source of happiness he was known to have said: "The greatest happiness is to vanquish your enemies, to chase them before you, to rob them of their wealth, to see those dear to them bathed in tears, to clasp to your bosom their wives and daughters." [11] A scientific support to this claim is made by mapping of Y chromosmes in Asia, showing a prevalence of 1/500 of all males being directly related to him. [12]
Actually, I particularly believe the last sentence in the above paragraph, not only about boys, but also girls in Central Asia, including Iran today.
There are more than a few Uzbeks, Tajiks, Afghans, Turkmen, and even those in both Republic of Azarbaijan, and the Iranian province of Azarbaijan, who could easily pass for half-blood Chinese, Monghol or Eastern Asian, because of their facial features, particularly their slightly slanted eyes (epicanthal fold). Some of them more obvious than others.
Whilst it could also be due to migration over the centuries, as well as weather conditions, I don't think migration or weather conditions are entirely responsible.