The answer will depend on two things, where in Asia and when in Asian you're referring to. You have to remember that the Mongol Empire was the largest the world has ever seen, and stretched from parts of South Asia all the way into parts of Eastern Europe. Due to this, their society was different depending on where you found them. Often, the Mongols would begin to adopt parts of the society of the areas they occupied (i.e. The Mongols in the Middle East/Central Asia began to adopt Islam as their religion, and reflected it in their culture, while Mongols who occupied China would take on more Chinese ways of life.)
As for mongol society that didn't involve this "borrowing of culture," The Mongols were nomadic peoples that lived throughout Central and Northern Asia. They were originally series of many different tribes, but eventually became united under Ghangis Khan through his policies of marrying off his children to other tribes rulers, simple alliances, and sometimes force.
The Mongol Empire also brought a lot to Asia (some people see their Empire as having been a benefit to the whole of Asia, really). Since most of Asia was dominated by the one empire, trading became much easier, with trade routes becoming more secure. Besides the spread of goods through trade, they also helped to spread ideas for the same reason. Several countries connected together by one Empire allowed the Mongols to spread ideas throughout their empire. In modern day you can still find Influences of the Mongol Empire in Asia, such as the continued practice of Islam in Central Asian and Western China, or the fact that they had helped several cities develop more quickly, thus cementing their place in history.
