Reading about the Mongol Empires ability to field and deploy vast armies going in all directions begs the question of how many people actually lived in the Empire to be able to field such armies, and how many of these were people that identified as Mongol.
Bonus: Were the Mongol armies consist primarily of Mongols or were they mainly made up of people from captured regions?
Sorry about the English.
This is only applicable for China.
The Mongols in China were notorious for their poor record-keeping, so the Yuanshi does not state the population of 1264. In 1235, Northern China had 873,781 households with a population of 4,754,975. In 1260 when Kublai ascended the throne, the Yuanshi states that the number of households had increased to 1,418,499. Assuming then that an average household had 5.4 people (based on the 1235 numbers), then the population in 1260 would have been around ~7.6 million.
The number of Mongols would have constituted a very tiny percentage of this. It has been estimated that the total number of Mongols and semu ren (Central Asians) households in China after the conquest of the Southern Song in 1279 was 400,000, only 3% of the entire population. By contrast, Southern Chinese households made up 82% of the population (Elizabeth Endicott-West Mongolian Rule in China, 86).
The Mongol army, then, would have been composed mostly of Chinese troops drawn from the former Jin territories of Northern China. The Mongols simply didn't have the manpower to overwhelm the Southern Song. We also know this from the Mongols' military policies in China because they practiced mass conscription (簽軍, qianjun), where they would conscript all able-bodied Han males to serve in the military. The Yuan official Hu Qiyu noted that between 1252 and 1274, the Mongols engaged in mass conscription a total of five times.