Jin, Xia, and Song Dynasties

And you forget that they were not allied, or unified under one banner when faced by the Mongols. The Jin (Chinese simplified, 大金, literally Large Gold [Jin]) dynasty was composed primarily of Jurchen tribes, and was at war with the Song (Chinese simplified, 宋朝, literally Song Dynasty). The Xia empire (Chinese simplified, 西夏, literally Western Xia) occupied an area south of modern-Mongolia.

The Mongols (and Genghis Khan) originally set out to consolidate their power over Western China by first attempting to conquer the Xia - the Mongols followed the Yellow River up to the capital, Yinchuan (银川, silver river). The garrison of the capital was more than double that of the Mongol army, and, being one of the first few attempts at siege warfare, the Mongols actually failed to capture the city. It was through Genghis' sheer persistence that by diverting the flow of the Yellow River to flood the city and the plains that the Xia submitted to his rule.

The Mongols then launched their invasion on the Jin. They managed to defeat a number of large armies on the field, as their style of warfare bridged more upon moving (horses) rather than stationary fighting. They were still rather horrible at siege warfare, so the Mongols enlisted engineers from the Middle East and China to develop siege engines of war designed to destroy fortifications and capture strongholds. Through this, they were able to conquer the Jin, and with this, they acquired new warfare tactics and superior military technology.

After the destruction of the Jin, the Mongols (now led by Taizong, then Xianzong, then Kublai - I'm not too sure about their Mongolian names) turned towards the southern Song dynasty. What's interesting about this campaign is that, rather than winning through war, the Mongols won through the inner political turmoil faced by the Song. The central ruling power found it hard to deal with both the Mongol invasion and rebellions inside China - this, coupled with an incompetent emperor, a puppet state chancellor, and a fragmented court with different families vying for power meant that it was easier for Kublai to capture individual strongholds. However, it was still with great difficulty and substantial loss of manpower that he was able to found the Yuan dynasty.

Sources:

Genghis Khan and the Making of the Modern World - Weatherford, J.

Genghis Khan: Life, Death, and Resurrection - Man, J.

The Cambridge History of China: Volume 6 - Alien Regimes And Border States 907 - 1368 - Franke, H. & Twitchett, D. C.