Genghis Khan and his descendants launched numerous invasions of China, subjugating the Western Xia in 1209 before destroying them in 1227, defeating the Jin dynasty in 1234 and defeating the Song dynasty in 1279. They made the Kingdom of Dali into a vassal state in 1253 after the Dali King Duan Xingzhi defected to the Mongols and helped them conquer the rest of Yunnan, forced Korea to capitulate through invasions, but failed in their attempts to invade Japan.. Which did Genghis Khan and his descendants conquer first, Korea or the Song dynasty?