So, Jalaluddin, the son of the last Khwarezmid Shah, was fleeing the Mongol conquest of Persia in 1221, he asked for refuge with the court of Iltutmish, the Sultan of Delhi. However, Iltutmish refused to give Jalaluddin refuge, and in reaction, Jalaluddin ravaged the Punjab contryside - a little pain for the Delhi Sultanate, but not much, and Jalaluddin soon returned to Persia to try to reconquer it. Now, let's say Iltutmish allows Jalaluddin sanctuary in Delhi. Genghis Khan would pursue him in India, which means that he'd then be invading the Delhi Sultanate - most of North India. I think he'd be able to conquer it after a few difficult and bloody battles, but for someone as great at battle as Genghis Khan, I think it would be successful.
What would be some effects of this? Well, I think the Mongols would have fewer conquests in Europe due to their rule of India costing them resources. This could mean that Kiev remains the centre of the Russian states, rather than moving north. Furthermore, if these conquests follow the pattern of others, I think that India would be tremendously depopulated. IOTL, it was a densely populated region for a very long time (and still is), but it would be less-densely populated ITTL - if it follows the pattern of the Persian conquests, India will not regain its former population for many centuries. And once the Mongols break up, a "Delhi Khanate" could spring up in India. The Khanate could then conquer the Deccan, bringing Mongol rule south and reducing populations there as well. As for the religion the Mongols would go, Islam was not well-established in India. The Delhi Sultanate had only recently sprung up, and it had not converted many Indians. IOTL, Turks fled from the Mongols into India, which led to the numbers of Muslims in India increasing, and this is one reason "Khan" is a common Indian name. These Muslims then converted native Indians, increasing numbers of Muslims further, to the point that today, one third of the Indian subcontinent is Muslim. ITTL, these Turks would flee elsewhere, and the Muslim inner circle of the Sultanate would probably be massacred. This would mean that Islam is much less strong in India. When the Mongols decide to convert, my bet is that they'd convert to Islam after being approached by Muslim proselytizers but there's a chance that they'd convert to Hinduism and become members of the Kshatriya caste. The chance of the Mongols converting to Hinduism is not very large, but it exists.
Thoughts?