To address this question you also have to address how they got so big so quickly. I can't speak authoritatively about their spread into Russia and Persia, but I can at least address their push south.

Leading up to the establishment of the Yuan under Kublai Khan, Eastern Asia had multiple smaller states fighting against each other, in the context of picking up the pieces after the fall of powers in the region, most significantly the Tang. First, the newly established Song was at war with the Liao in an effort to take over regions that the Liao had taken from the now-fallen Tang. The Song was then fighting with the Tangut (Western Xia) for over a decade. The Liao is then broekn down by the Jurchen who form the Jin dynasty at this time. The Western Xia are still fighting with the Song and both are having their armies substantially reduced at this time. The Liao finally falls to the Jin and the Jin move on to the Northern Song capital, forcing the Song court to relocate in the South. Fighting continues, land changes hand. The Mongols arrive shortly after and wage war with the Western Xia followed by attacks on the Jin. Keep in mind this is after significant fighting on all sides involved before the Mongols attacked. Over the next few decades, the Mongols fight their way south conquering much of China and eventually taking the Song capital of Lin'an.

Had the mongols arrived some time earlier, for example when the Tang was at its height, they would likely had not been so successful. Their spread into China was in small part due to coincidence. That is, there was a certain degree of fortuitousness in the timing of their conquests south.

By the end, the Mongol Empire was split into different khanates. So again I'm just addressing that which was the Yuan Dynasty.

Starting around 1300, the Yuan saw revolts, floods, plagues, more revolts, civil war, famine, loss of control and the eventual fall of the dynasty. The Yuan was ended the same way most dynasties have ended in China, with natural disasters and a loss of control. One book I've read on this topic has attributed their fall to a general inability for a nomadic society to effectively rule large areas as a stationary government, since their strength in battle comes from mobility, which is lost when you're now managing grain stores and large cities. I can't say if that's an accurate assessment. What I can say is that, even if the Yuan were quite adept at managing a state, nature and the native Han population were not working out in their favour, and no amount of civic management would have prevented the end of the Yuan.

I'm eager to hear what others have to say about other parts of the empire, as well as a more qualified contribution of the big picture.

You need to differentiate between Mongols and Mongolia. The Mongols quickly conquered a huge land empire. In the days before telephones and trains, it was hard to administer such a huge empire, so it is not surprising that it broke up.

However, Mongols continued to rule parts of their old empire for centuries. The Chagatai Khanate existed in central Asia until the late 17th century, the Golden Horde until the 16th century. Offshoots such as the Temurids ruled huge land empires until the 1500's, the Mughals in India up to the 1850's. Through much of history in this period, being able to trace descent from the Great Khan vastly improved your chances of leading men, whether as emperor or king or tribal warlord.

The Mongols were few in number, and they quickly adopted customs of the lands they conquered and became part of them. Today, you find Mongolian features among a whole range of populations formerly conquered by the Mongols, so you can see they never left, they are part of the gene pool now.

So if you consider the Mongols, they remained alive and powerful for a very long time in different parts of the world. But if you consider Mongolia, it's a sparsely populated semi-desert, and after Genghis Khan's empire broke up it became less and less important to the Mongols, who became Chinese, Indians, Persians, Afghans, Central Asians, whatever.

It is commonly observed that many of those who conquered and expanded rapidly, could not effectively govern and hold onto their territories.

Military power can conquer, but it takes culture, economy, technology to hold onto territories and solidify rule.