Between 1251 and 1254, Mongol armies subdued the Goryeo Kingdom of Korea, though not without drama. Under military pressure, the Korean king sent them a hostage who was supposedly his son, but it turned out to be a stepson not of the royal blood. This was apparently a deliberate deception, an attempt to cheat the system. Möngke Khan was furious; he held the entire court responsible and ordered the land to be razed. The Goryeo court fled to islands, reasoning that Mongols didn’t sail. But Mongols could order and pay Koreans to sail ships for them. Now with naval experience, the Mongols finally retreated with the actual crown prince as hostage and the land thoroughly cowed (that is, burnt, starved, destroyed). The Korean kingdom served Mongol officials after that.
China’s inland Dali Kingdom in modern Yunnan Province was next on the list. Möngke’s brother Kublai conquered the capital city of Dali then sent a column south, where there was another route through Vietnam to get at the Song Dynasty. The Song Dynasty should have fallen to Genghis Khan long ago, by Mongol reasoning; but it had retreated south, leaving only North China open. Vietnam was uncooperative, so it was conquered; Hanoi was sacked and occupied. But as before, the climate of Vietnam was terrible for Mongol health. The Tran dynasty accepted Mongol overlordship and paid tribute, so all of the Mongols but a few unlucky tribute officials left behind could race north to the dry, cool air again. (Kublai had wisely gone no farther south than Dali.)
After a few years, Möngke Khan was looking for a conquest zone that was not tropical, and apparently his eastern limits had been reached for now (Japan was out of their range). To the west, he’d received the submission of western Armenia and Antioch, the Sultanate of Rum, and the Emirs of Aleppo and Mosul, almost without lifting a finger. The last Mongol invasion had made it amply clear to these rulers that it was much, much better to be an ally of the Mongols than a lone, proud, hold-out waiting for the siege to arrive.
In 1255, following the great census, Möngke named his brother Hulegu as Ilkhan of this southwestern region. This Vice-Khanship wasn’t defined by borders but by its dynastic range. It was defined as being for the family of Hulegu; they were allowed to make it much larger as long as they didn’t encroach on the lands of Batu’s lineage based in Sarai or Kublai’s lineage in China. The Ilkhanate included Afghanistan, Iran, Armenia, Georgia, Pakistan, and India, really as much as Hulegu wished to conquer before he’d get done in by the climate.
Hulegu had a special mandate to subdue remaining Muslim states. This meant the Lur people of coastal Iran, the Nizaris of Iran and Iraq’s mountains, and the Caliph of Baghdad. Everyone else had been conquered or had submitted. And so the Third Mongol World War began.