After the death of Kublai the Yuan dynasty declined in power as officials became corrupted and other khanates in the west distanced themselves from central control. In the end they were overthrown by the Mings and they retreated to Mongolia. In the west the Golden Horde allied with the Mamluks and attack the IlKhans (Persia). The Il Khans sent Rabban Sauma, a Chiense nestorian monk to negotiate alliance with Christian Europe, but the Europeans were too busy fighting amongst themselves to listen. In the the Il Khans adopted Islam and became assimilated into Persian-Islamic society. The Il Khanate fragmented in the mid 14th century and the remnants were conquered by Timur.
In central asia the Chagatai Khanate tried to maintain steppe tradition in the face of growing Islamic assimilation. There were succession crisis and then they were conquered by Timur of Samarkand, which is a setback the Chagatai never recovers from. The Chagatai Khanate survives until the 15th century under Turkic/Islamic rulers until they fragmented under pressure from Uzbek and Kazakh attacks.
Of the four post Genghis Khanates the Golden Horde fared best of all. The empire were made up of mostly Turkic and Mongol people and grew rich from combination of plunder, tributes and trade. In the mid 13th century the rulers converted to Islam and Sarai became a large Islamic center of learning and trade. The Golden Horde remained the most powerful state in Europe until the late 14th century, when civil wars, weak rulers and the Black Death epidemic took its toll. Russian principalities became increasingly rebellious. Things looked a little brighter after Khan Toqtamish defeated the Russians, but then Timur attacked and sacked Sarai. The Golden Horde declined rapidly and eventually fragmented into several Turkic khanates eg Astrakhan, Crimea, Kazan etc which would later be defeated by the Russian Empire.