I could imagine a world where Vietnam is incorporated into China, and where the Chinese control the flow of Indian Ocean trade, collecting tribute from states all the way to the east coast of Africa. In such a scenario, you would see hybridising of cultures. Perhaps a south India that more closely resembles OTL's southeast Asia?
Thing was, even before the Ming the Chinese tried to avoid culture hybrization and were quite good at assimilating native populations into China-- I mean, everybody thinks of Yunnan as solidly Chinese now, but Chinese conquest and occupation of the area only started in the 1300s (I think).
The trick was to promote Mandarin langauge and culture through regulation and examination. If you wanted to be any part of the Chinese Civil Service you had to not only be able to read, write, and speak Chinese, but also display a solid knowledge of Confucian texts and Chinese culture.
As for occupying Kashmir, that'd be tricky for any Chinese dynasty pre-Qing. The Tang dynasty just
barely bordered what's known as Kashmir, Mongol expansion during the Yuan period likely wouldn't result in anything recognizably Chinese, and the rest of the dynasties generally didn't expand further west than Sichuan. A better route for China expansion might be through Yunnan and heading southwest from there, through parts of Burma and Bangladesh.