If you're trying to figure out China's next move in Hong Kong or how India will proceed in Kashmir, here's a clue: Follow the logic of an empire. China and India each inherited control from the British Empire, and are following a script that could have been written a century or more ago. Both governments probably have more legitimacy among their subjects than the British Empire did, but that's beside the point when it comes to their reasons for acting today.
Start with China, which got Hong Kong back from the United Kingdom in 1997. China promised "one country, two systems," an arrangement that was supposed to allow a common-law-style judiciary to continue operating in the former British colony. Yet it's not as though Hong Kongers enjoyed democratic self-government under the British. The handover of Hong Kong was the exchange of one empirical sovereignty for another. The government of China was just much closer at hand, and had a stronger traditional claim to the territory.
No imperial government can afford for its subjects to challenge its ultimate ability to lay down the law. When the anti-government protests began in June, the impetus was a proposed Hong Kong law that would have allowed extradition to China for some crimes. That was arguably not inconsistent with "one country, two systems." The protesters claimed to the contrary.