A vaguely similar question came up before. I'll copy my response from there because I think it should suffice and I'm way too sleepy to give you a unique writeup right now.

Numerous Christian communities resisted the "Nicaean regimentation" of Christianity in the 4th and 5th centuries. Those beyond the reach of the Roman Empire, mostly to the east, could get away with maintaining their own, separate doctrines and practices. Christianity was originally very diverse in views on the divinity of Jesus, Mary, and the Trinity, so while these views were somewhat standardized in Rome, those who believed otherwise might leave for the east to follow their own form of Christianity. The political infighting within the early Church, which involved a few high-profile exiles, encouraged a little more fleeing.

The "Nestorian heresy" emphasized the humanity of Jesus, and found refuge in the Zoroastrian Sassanian Persian empire. The Sassanians tried to use these Christians to prop up a rival church to undermine Rome. Persia had ties to central Asia and even China because Sogdia was part of their empire, and Sogdians were some of the most active merchants on the "silk road". Most Sogdians were Zoroastrian, but Nestorian Christianity became the largest minority religion in Sogdia.

Nestorian Christianity became (perhaps surprisingly) very popular among Turks, early in their history when they dominated the central Asian steppe. There are stories of mass conversions among the Turks after witnessing miracles. Some Sogdians carried Nestorian texts into T'ang China, translated them into Chinese, and obtained imperial permission to preach Christianity in China. By the 1100s there was even a significant Nestorian population in Mongolia. Some of Chenghis Khan's generals were Christians!

For the most part Nestorianism declined in popularity after the 900s, and it had never caught on in China. Taoist and Confucian scholars convinced the emperor to suppress Christianity after a rebellion by a Sogdian general, since the Chinese saw Christianity as a "barbarian" religion. Nestorianism also declined in Iran because of the Islamic Arab conquest of the Sassanian Persian empire.

For central Asian peoples, Mongols and Turks, Islam simply became a better choice for conversion for social, economic, and political reasons. Islam dominated the trade routs, empires, and populations to the west, and wasn't associated with rebellious barbarians in the east. Despite the popularity of Nestorianism among Mongols, it simple could not compete with Islam once the Mongols started building an empire in the 1200s.

So Christianity did spread east, but it was simply beat out in the competition with other religion. India was already the "battleground" for Hinduism, Buddhism, and Jainism, all of which were older than Christianity and more entrenched. Islam made its way into India entirely because of invasion, trade, and migration. Christians were too distant to do any of that until the early modern period.

Source: class notes from a "cultures and empires of the Silk Road" class with a well-regarded professor. Funny thing, journal reviews for that professor say the exact same thing his students do about him: dude knows his stuff but couldn't possibly be more dry.

It did actually spread much further than that—here's a map of the (Nestorian) Church of the East at its maximum extent in the Middle Ages.

As to why it declined after that point, I’d guess it was mostly competition with Buddhism, which had a similar appeal.