Which of these cultures ruled over much of Central Asia for more than two centuries?

You didn't provide any possible answers, but the correct answer is surely the Mongols.

The reason for this is that the Mongol Empire expanded accross multiple continents and spanned over two centuries which is something no other empire at the time could achieve - and only the Mongols fit that description for that reason.

The Mongol Culture

It existed during the 13th and 14th Century., the central Asia was heavily affected by mongol nomadic Way of life. In this Culture, Tengrism was dominant belief system characterized by Shamanism, animsim, totenism, and Ancestor Worship