The Mongols as a people were split into several large tribes across Asia - they were united by Temujin (or Genghis Khan) and for the first time in a while a unified military force was established under his rule. In east Asia at the time, China was split into 3 dynasties - Jin, Xia, and Song - and they weren't allied nor unified when the threat of the Mongols arrived. This in turn meant that the Mongols could use the help of another dynasty to focus the one immediately head, instead of having to deal with all of them at once.

No doubt you've heard of the story where some person diverts the flow of a river over a city and wipe that from history. The Mongols were inexperienced at siege warfare, and so when they came upon the first heavily fortified garrison they attempted to flood out the city and its plains. This failed, but the continual pressure from the Mongols coupled with the lack of help from the other Chinese dynasties soon forced the Emperor of the Xia to submit.

What was special about the Mongols was that, unlike their contemporaries, their style of warfare was more upon horseback, and especially, archery on horseback. They would essentially surround the enemy host with horses and fill them with arrows before going in - their superior mobility allowed them to do this, whereas their opponents, not so much.

I also wouldn't say that they came and left in a short period of time - they did establish the Yuan dynasty in China that lasted close to a hundred years, and they did impart many of their attitudes and approaches to other cultures.

Sources:

The Mongol Conquests in World History - May, T.

The Mongols wound up being the most successful Steppe Confederation in terms of sheer landmass... and they're also burned most deeply into the world's memories because they impacted both the Muslim Caliphate and Medieval Europe so dramatically - destroying the former and ensuring the latter knew just how far down the social and warfare totempole it actually was.

In fact, however, the Mongols were far from the first confederation to strike terror into the hearts of Asia in general and the Chinese specifically. When the tribes of the Asian Steppes managed to put aside their own feuds long enough to turn their focus outward, they were every bit the terror of Genghis, and far, far earlier than the 13th century.

The Xiongnu Empire

The earliest steppe confederation that is known to us were the Xiongnu (also romanized as Hsiung-nu). We don't know their real name since they were a preliterate society, and so we only have the name their prime adversary gave them, meaning "fearsome slaves."

In spite of the moniker, though, the Xiongnu were rarely subservient to the Chinese - and far more often the opposite was true. They became a raiding group of sufficient note to warrant inclusion in Chinese annals as early as the late 3rd century BC, when the proto-khan (then called chanyu) Modu (or probably more like “Batur” in proto-Turkic) seized control of his father’s tribe after killing him and united the surrounding peoples in 209 BC.

This unification came just prior to the collapse of the hugely important, but hilariously short-lived Qin Dynasty in 206, throwing the just-unified China into a state of civil war as the warlords of the Han and Chu states vied fill the power vacuum, and in the process severely weakened the state, especially along the border regions between 206 and 202 BC. This meant that by the time the King of Han claimed victory and created the new Han Dynasty and himself Emperor Gaozu… the Xiongnu to the north were already raiding, pillaging, and looting to their hearts content in the far wealthier farmlands south of the Ordos Loop of the Yellow River. This would result in a brief war between the two which would result in Gaozu’s swift capture, and the Empress herself forced to buy him back by promising to the Chanyu that the two empires would enter into a “marriage alliance” system (called heqin), which was far closer to a tributary-status for China than any real alliance. It would be forced to pay annual “gifts” to the Xiongnu Empire as well as marry noble girls to the northerners’ chieftains.

The Xiongnu Empire would exist in this state for alonst three quarters of a century, until Emperor Wu of Han got tired of this constant embarrassment and – as part of his massive series of territorial expansions – renewed all-out war with the Xiongnu in 129. Armed with a new breed of horses from Central Asian kingdoms and a far better idea of the empire he was facing off against due to an epic inland exploration odyssey, Wu’s armies managed to defeat and break the Xiongnu Empire’s hold on both the Ordos Loop, and even more importantly their stranglehold on the Western Corridor out of China and into Central Asia around the Taklamakan Desert – the route that would become eventually the Silk Road.

With the tables turned, the Xiongnu rather quickly devolved into infighting – lack of new conquests would ultimately be the undoing of many of the Steppe Confederations… with no new blood, the uneasy coalitions of tribes would turn on each other once again. Ultimately they’d be partially absorbed into Han China and sinicized. That said, they would play a major part in future Chinese and Mongolian history to come…

Notes:

Fan Ye, Hou Hanshu, vol. 89

Sima Qian, Shiji, vol 110

Xianbei

The steppe peoples continued to be a nuisance to the Chinese, but it wasn’t until the Middle Kingdom once again fell into civil war and broke apart once again that a new manifestation of a centralized Mongol-Turkic state would take place. That would occur at the outset of the 16 Kingdoms Era of China – which had itself come about after the epic (and epically devastating) 3 Kingdoms six-decade conflict of the 3rd Century CE.

The short-lived victor of the Three Kingdoms, the Jin Dynasty, rapidly fell into corruption and ruin, resulting in an uprising led initially by the Xiongnu (told you they’d be back). Once again, they (now ensconsed within China itself, thanks to relocation programs aimed at de-barbarian-ing them) headed a multi-ethnic force, mostly composed of various tribes of the Xianbei People.

With China’s population lowered by more than half thanks to the 3 Kingdoms conflict, the north of China was in a very poor position to defend itself… and even moreso since in it dying throes, the Jin had turned on itself and begun massive purges and slaughters against its own people (a factor that led to the Xiongnu rebellions in the first place). This would result in the Jin losing their capital, and fleeing south to what is now Nanjing, establishing the Eastern Jin, and leaving the devastated ruin of the north to the Xianbei Barbarian Empire. Unfortunately for them (and hence the Era name 16 Kingdoms) the Xianbei and the other ethnicities that had so rapidly managed to give the boot to the ethnically Han population and monarch quickly devolved into (yep, you guessed it) terrible infighting that would last more than a century from 304 until 439.

Tuoba Xianbei (Northern Wei)

In 399, the warlord Tuoba Gui established the state of Northern Wei and himself as its Emperor Daowu. It wouldn’t be until the 439 that the northern bloodletting would finally come to a halt when Daowu’s grandson, Taiwu, finally crushed remaining resistance and united whole of Northern China under his iron-fisted rule.

Northern Wei would never control must south of the Yellow River, but it retained its lock on all of Northern China for more than a century, until it would finally succumb in 535. During its tenure, its Tuoba Xianbei ruler became so sinicized as to be almost indistinguishable from their Southern Han-ethnicity counterparts. They made many cultural, technological, and religious advancements that would remain hugely important to Chinese civilization long after the Xianbei became irrelevant.

And ironically, they faced a very similar threat as their souther foes: barbarian invasion! While the Han Chinese of the south were ever concerned about Northern Wei sweeping them off the map, Wei itself had to deal repeatedly with yet another Northern Steppe people: The Rouran Empire, which would exist as a political and military force until 20 years after the collapse of Northern China in 555.

Notes:

Wei Shou, Wei Shu (Book of Wei)

Li Daishi, Beishi (History of the Northern Dynasties)

Sima Guang, Zizhi Tongjian (Reflections on Governance), esp. vol. 106-152

Jurchen (Jin)

Northern Wei would be the last time in a while the northern “barbarians” would do much than pester the Chinese (which you might note was becoming a rather “catch-all” term for all peoples who walked and quacked enough like a duck to pass muster). We can’t really call them Mongolic, but the Jin Dynasty of the 12th century that would sweep the Northern Song down to the South (again!) were nevertheless a steppe people, this time from Manchuria: the Jurchen.

Like the Xianbei before them, the Jurchen came screaming down on horseback fighting in a manner that was all-but-impossible to counter. Once situated in power, however, they adopted Chinese language, mannerism, and fighting techniques… enough so to the point that when some upstart called Temujin united the Steppe Tribes once again in 1206 and was proclaimed Khan of the World, it would be Jin China that would face his wrath and its own destruction first.

Notes:

Tao Jing-shen, The Jurchen in Twelfth Century China (1976)

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If you’d care to listen to more about ancient China, we’re actually in the middle of the Northern Wei period (called the Southern and Northern Dynasties) at The History of China Podcast.

Genius Kahn was the first to really unite all of the Mongolian step people in a meaningful way, so that contributed to their emergence on the world stage. Other than that their sudden rise is probably tough to explain.

Their fall is attributable to issues specific to the Mongols. Specifically, they had no concrete succession plan. After the death of the second great Kahn, Ogedei, the Mongols always had a very difficult time choosing a leader. This led to the partitioning of their empire.

Your question seems to get at a different issue though, which is why did the east produce so many great horse archer armies for a while and then suddenly stop? One reason (I'm sure there are many) is the rise of gunpowder.

To be a great horse archer like the Mongols took a lifetime of training. A king raising an army could not simply pay for more training to turn his soldiers into true horse archers, so on certain battlefields non steppe people were kind of helpless.

Early firearms were inferior to bows, but they took just a few weeks, rather than decades, of training. This allowed for an army relying on firearms to replace losses over time, something that the steppe people had trouble doing. Finally money could be used to defeat the horse archers.

This is just one of many reasons I'm sure, but it is one I know a little bit about. I hope that helps.

Also I'm on mobile so I hope there aren't any mistakes.

I would say the premise is wrong.

I don't know much about the Golden Horde but they definitely left their mark on Russia. And so did the Mughals on India.

In China even after the overthrow of the Yuan, the Mongols continued to heavily influence foreign policy until the high Qing. For example, Zhenghe's voyages were cancelled and the current Great Wall were built both due to a resurgence in Mongol power.