In east Asia at least, these civilisations both fought with China and established peace with the dynasty of the time, whether through gifts or marriage. The main reason why they wanted in was because eastern Asian regions were more fertile, and in general a less harsh place to live in comparison to the lands of Central Asia. Especially with the Mongols, as a nomadic civlisation, they had little in the way of agriculture and infrastructure - they weren't 'settled' in one place, rather that they were pastoralists. Imagine a shepherd with a flock of sheep (as an example of an animal) that moves from one place to another.

The Mongols needed new grasslands to allow their herds to graze, and if you look at this image of the Eurasian Steppe you'll realise why they wanted to expand. These were bordered by China to the East, and so in turn were another incentive to expand into China. I can't say much for the other tribes, but this was a large reason of why the Mongols at least wanted to conquer east Asia.