Sorry, not if you were Han (Chinese). Their rule was brutal, but short lived. Where did you get this Ghengis Kahn's decedents ruled into the 20th century from?
As for the warm climate fuel the Mongol expansion into the West, too early to speculate at this point.
I think he meant 19th century. The Moguls of India were, at least in theory, his descendants and they weren't kicked off the throne in India until the Mutiny of 1857. Though their rule had been pretty theoretical for quite some time by then.
Also pretty much all the Central Asian rulers down to their conquest by the Russians were his descendants.
About 8% of the population of a rather large area of Asia may be descended from him.
http://en.wikipedia.org/wiki/Descent_from_Genghis_Khan#DNA_evidence_-_The_Genghis_Khan_Effect
Actually, looks like the Qing Emperors of China were all his descendants, with their empire lasting till 1910.
In fact the Japs put the last of this line in as the puppet ruler of Manchuria, which lasted till 1945.
"Ghengis Khan and the Making of the Modern World" by Jack Weatherford.