Edit: Chris has something there, brief as it is. Rome expanded successfully because they incorporated the religions of those they conquered. Hinduism and Buddhism were growing as the Chinese and Indian cultures were growing. Rome was dealt a blow when a guy named Jesus started causing trouble, and a bunch of people took up after him. Maybe you can find something similar in the other two.
This may not be much help, but their falls may be a lot more similar than you might think, being as how the Western view of World History is really veiled as European history. Consequently, I know a lot more about Rome than China or India, so like I said, not too much help.
There is always the theme of the giant, expanding power becoming too big to sustain itself. I have trouble wrapping my head around how the governments of China and India can manage a billion people each today, let alone when the fastest means of communcation was on horseback.
