They controlled on of the largest land empires in history and conquered the Islamic world and destroyed the Song dynasty in China, and conquered all the way to Eastern Europe. What stopped them from taking India as well?
Well, fun fact, they kind of did.
I won't go into too much detail, but the Mughal Empire, who ruled much of India from the 16th to the 18th century, were in fact descended from the Turco-Mongol people of Central Asia.
In fact, the rulers of the empire claimed direct relation to Genghis himself. While I'm no expert, i believe the Mughal's conquering of India is considered connected to the Mongol conquests in a similar fashion to other break away elements of the larger Mongol invasions.
For some further reading,
Richards, John F. (1996). The Mughal Empire
Gommans; Jos. Mughal Warfare: Indian Frontiers and Highroads to Empire, 1500–1700
Dale, Stephen F. The Muslim Empires of the Ottomans, Safavids and Mughals