The Mongols didn't do the impossible, as they were not in fact the only foreign rulers of China. When Chinggis began to invade what is now China in 1211, China's northern half was ruled by the Jin Dynasty, established by the Jurchens of Manchuria, to the east of Mongolia. Other users with more familiarity with this period of Chinese and Mongol history can probably say more in detail about the period (/u/lordtiandao, /u/krishaperkins or /u/cee2027), but in short regarding the Mongol conquest, it was helped along immensely by the fact that China was in fact divided between three states – the Chinese-ruled Southern Song in the Yangtze and Pearl River basins, the Jurchen-led Jin Dynasty in Manchuria and the lower Yellow River basin, and the Tangut-ruled Western Xia on the upper Yellow River. The Mongol invasion was therefore hardly coming at a time of strength and unity for China.

Now, I'm no specialist on the Yuan so I will avoid making much commentary on China during the Mongol conquest, but instead I'll redirect your attention for a moment to the other time China was fully ruled by a foreign people. Remember the Jurchens up in the north? Well, they came back, and in fact they ended up ruling China far longer than the Mongols did – 267 years from 1644-1911, as opposed to the Mongols' 97 years on the throne from 1271-1368. At the turn of the seventeenth century, one Jurchen chieftain, Nurhaci, began to reunite the various Jurchen tribes into a so-called 'Later Jin Dynasty', and the Jurchens once again challenged the rulers of China, launching raids against the Ming Dynasty that had displaced the Mongol Yuan. Among other things, Nurhaci established the Banner system, whereby all Jurchens were organised into four army units identified by the colour of their flags – the Yellow, Red, White and Blue (later doubled by splitting each into a Plain and a Bordered Banner.) While Nurhaci himself died in 1626, his son, Hong Taiji, continued his efforts, and in 1635 renamed the Jurchen people to Manchu, and in 1636 the Later Jin became the Qing. Although Hong Taiji achieved some success raiding across the Great Wall, he nonetheless died in 1643 without taking control of China proper, something that fell to his son Fulin (a.k.a. the Shunzhi Emperor), said son's regents Dorgon and Jirgalang, and his grandson, Xuanye (a.k.a. the Kangxi Emperor). While your more specific question is directed at the Mongols, a look at how the Manchus conquered China would also provide some interesting perspective against which the Mongols can be weighed (as and when an answer does come up).

The first key thing to bring up is that the Manchus did not breach the Great Wall in 1644, but were instead allowed in. As in 1211, China was a divided country, with a peasant rebellion based in western China contending with the established Ming Dynasty with its capital in Beijing, and in 1644 Li Zicheng, leader of the rebels, besieged and captured Beijing, driving the Chongzhen Emperor to suicide, and declaring himself the Yongchang Emperor of the Shun Dynasty. Wu Sangui, commander of the army based at the Shanhaiguan Pass, finding himself stuck between rebel-held Beijing on one side and the Manchus on the other, sided with the latter, believing that they would at least preserve imperial institutions as opposed to the radical agenda of the peasant rebels. The Manchus drove Li from Beijing after barely two months on the throne. Throughout their conquest of China, the Qing would continually insist that they were there to suppress Li's revolt and, in the apparent absence of legitimate Ming claimants (those in the south being mere pretenders in their rhetoric), restore order on their own terms.

Defection and collaboration was the hallmark of the conquest, and although the Manchus were not universally popular (especially so after Dorgon's 1645 queue edict forced all Chinese men to shave their foreheads and adopt the Manchu queue), they nonetheless got support where it counted. Wu Sangui's defection was just one of many. As early as Nurhaci, a mixture of impressed prisoners and voluntary defectors from the Ming had been incorporated into the Jurchen/Manchu organisation, and these so-called 'military Han' (Hanjun) made up a sizeable portion of the Banners, such that in 1631 Hong Taiji split them off from the Manchu Banners to form eight 'Han Banners', lower in the order of precedence but nonetheless a privileged group relative to the rest of the Han Chinese population. The Mongols, too, had some Chinese defectors in this way, with two Chinese tumens (around 20,000 soldiers) during the invasion of the Southern Song. Not all Han Chinese defectors became Bannermen, however, most prominently Wu Sangui and his army. Yet there were both be older defectors and newer ones – Shang Kexi, based in Liaodong, had gone over in 1634 when the Manchus took over that province, while in 1646 the Fujian admiral (and ex-pirate) Zheng Zhilong and his subordinate Shi Lang went over, betraying the Southern Ming Longwu Emperor to the Qing, who captured and killed him in October. While initially ennobled under the Qing, Zheng rapidly fell from favour and was executed in 1661, due mostly to the intransigence of his son, Zheng Chenggong (more commonly known as Koxinga, the Hokkien pronunciation of his title of Guoxingye), who had continued to declare for the Ming and had control (both direct and otherwise) over many of the major port cities south of the Yangtze, such as Fuzhou, Qianzhou, Xiamen and Wenzhou, and even once made a major foray up the Yangtze in an attempt to capture the city of Nanjing. (In 1661-2 Koxinga sailed to Taiwan and expelled the Dutch from their colonies in the southern part of the island, dying shortly afterwards.)

Collaboration was also a necessary part of consolidation. The Qing inherited the civil service system of the Ming (itself an inheritor of the Mongol system) and staffed to a great extent with Han Chinese scholar-officials, although right down to the end of the dynasty, the Manchus, who made up a mere 1% of the population (and this is if we take 'Manchu' to include all Bannermen, including Mongols and Hanjun) remained disproportionately overrepresented at all levels of government. Even so, the southern provinces where the Ming had tried to hold out were nonetheless deemed too unstable for direct Manchu rule, and until 1681 were divided into vassal states called the Three Feudatories, headed by the early defectors and their descendants – Wu Sangui ruling the southwestern inland provinces of Yunnan and Guizhou on the border with Burma and Vietnam, Shang Kexi ruling the southern coastal provinces of Guangxi and Guangdong, and Geng Zimao ruling the southeast coastal province of Fujian. This was not wholly perfect, of course. Wu and Geng revolted in 1671, and Shang Zixin's son in 1680. However, the Qing, having now also established the Green Standard Army for non-Banner Chinese, were able to crush the revolt of the feudatories in 1681 and extend direct rule over all the old Ming lands. The same year, Koxinga's son Zheng Jing died, leading to infighting among the leaders of the Taiwanese loyalists, which, combined with a severe storm that destroyed many of the Taiwanese ships, allowed the newly-built Qing fleet to defeat the Taiwanese near the Pescadores (Penghu Islands) in 1683 and force them to surrender. Even then, co-optation remained viable, as Taiwanese veterans would be recruited to fight the Cossacks on the Amur River during the Siege of Albazin (1685-7) – and this bears some comparison to Wang Jingwei, a senior figurehead of the Nationalist Party, who having supported Chiang Kai-Shek's call for war in 1937 became head of the pro-Japanese collaborationist regime in 1940.

So it must be said that the Mongol conquest was hardly wholly unique. Once again, apologies for being unable to address the question directly, but I hope it's at least given a little glimpse of how a foreign conquest could be carried out.

Bibliography:

• ed. Willard J. Peterson, The Cambridge History of China, Volume 9, Part 1: The Ch'ing Empire to 1800 (2002)
• Jonathan D. Spence, The Search for Modern China (2nd ed. 1999)
• Jonathan D. Spence, Emperor of China: Self-Portrait of K'ang-Hsi (1978)
• Tonio Andrade, The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History (2016)
• Tonio Andrade, Lost Colony: The Untold Story of China’s First Great Victory over the West (2011)

Your statement is false.

The Qing dynasty, which ruled all of modern China from 1683 (with the Qing conquest of Taiwan) to 1843 (with the surrender of Hong Kong Island to the United Kingdom following the Opium War) and remained in power over almost all of China until 1911-1912 was of Manchu ethnicity, and the Manchus had their own language, script, culture, religion, traditions, and homeland (Manchuria). Manchuria until the 17th century was not part of China except in the loosest sense of being part of the Chinese Ming Dynasty’s tributary system – and since the Ming tributary system included states as far way as Malacca and modern Indonesia that are not regarded by modern historians as part of Ming China proper, the mere participation in the Chinese Ming Dynasty’s tributary system is not evidence that a place was part of China.

The origins of the Qing dynasty are, strangely enough, closely linked with the formation of the Manchu ethnic identity. Nurhaci, as a leader of the Aisin Gioro Jurchens (not Manchu!), from 1582 began warring against other Jurchen clans and tribes in order to unite them under his rule, For much of this time, he and his Jurchens were part of the Ming tributary system, even when, in 1616, he proclaimed himself to be the Khan of the Great Jin – sharing a name with the earlier Jurchen Jin dynasty that had conquered northern China.

This changed in 1618 when Nurhaci formally renounced his status as paying tribute to the Ming Dynasty and conquered the last few Jurchen tribes that were not under his control but allied to the Ming. As a result of this, Nurhaci found himself at war with the Ming, but was able to succeed against their armies, not merely defeating Ming forces but conquering Ming territory in the north (the Liaodong peninsula) that his forces secured enough that he was able to move his capital south – by 1625, his capital was at Mukden.

This southern expansion brought Nurhaci and the Jurchens into greater contact with the Korchin Mongols, whom Nurhaci allied with against the Ming Dynasty and largely integrated into the Jurchen state through marriages. Those few Korchin Mongols who were not willing to be so allied, he conquered and integrated into his state and army. The result was that ethnic and linguistic Korchin Mongols became an integral part of Nurhaci’s Jurchen state and army.

Nurchaci created, based upon the Mongolian script, a new script to write the Jurchen language. He also created a system of mixed civil and military administration to govern his territories and integrate other ethnicities within the state and its army – but always with the Jurchens in ultimate control. In later decades, this system would be refined into the Eight Banner system.

This banner system, even in its earliest form, was useful as a way to systematically integrate Mongols. A similar approach, not yet formally integrated into the banner system, was used to integrate Chinese people, whether as captives, occupants of newly conquered lands, or defectors from the Ming dynasty, into the Jurchen-controlled state and army that Nurhaci was creating.

In January 1626, however, a Ming general named Yuan Chonghuan, using Portuguese cannons, defeated Nurhaci’s army and wounded Nurhaci, who died later that year and was succeeded by his son, Hong Taiji. This did not stop the defeats, and in 1627, Yuan Chonghuan, again using Portuguese cannons, defeated the Jurchens.

Hong Taiji, however, was not overthrown, nor was his state destroyed, and he did not give up.

At this stage, corruption within the Ming state granted the Jurchens a reprieve. Yuan Chonghuan was criticized by a faction within the Ming State for not being swift enough in his campaigns against the Jurchens and for executing a skilled military commander (Mao Wenlong) for excessively casual attitude towards protecting civilians from Jurchen attacks (basically using them as human shields according to his critics). But Mao had been able to build up great influence within the Ming court by bribing eunuchs and officials, so that Mao’s execution, in combination with Yuan’s inability to win a decisive victory against the Jurchens (only a few minor victories after much indecisive campaigning), meant that Yuan’s corrupt enemies were able to get him forced to retire. However, in 1628, the Ming Chongzhen Emperor personally intervened and restored Yuan to military command over Ming military forces in the north-east, after which Yuan devised a campaign to decisively defeat the Jurchens and reconquer the Liaodong peninsula. In 1629, Yuan was assured by the Emperor that he would support whatever military decisions Yuan would make, and Yuan went forth to campaign against the Jurchens.

However, the Jurchens had taken advantage of this lack of Yuan’s military skills to do their own military campaigning and reforms. They conquered more tribes, integrated more Chinese and Mongol troops into their armies, and forced Korea to pay tribute to them. Thus, when Yuan went north to fight against the Jurchens in 1629, he faced a larger army than he had in 1626.

However, the Jurchens refused to wage war as Yuan had expected. Rather than attacking where Yuan had expected them to attack, they went elsewhere, broke through the Great Wall of China, and advanced upon Beijing. Yuan learned of this and, leading an elite portion of his army south, fought against and defeated the Jurchen army on the outskirts of Beijing. However, this battle did not destroy the Jurchen army, which was able to withdraw in good order. Yuan, however, was condemned for not doing more to defend Beijing (the Ming dynasty's capital), and some eunuchs accused him of having allied himself with the Jurchens. So, in January 1630, Yuan Chonghuan was arrested, and in September of that year he was executed.

Hong Taiji, however, recognized that Yuan Chonghuan’s victories had been in great part due to the use of cannons, so he secured his own cannons and in 1634 even created an artillery corps using Chinese cannoneers equipped with the latest types of artillery made within his state by defecting Chinese gunsmiths. In 1635, he formally introduced the banner army system, granting Mongols their own banner and hierarchy under Jurchen command. In November 1635, he ordered that the united Jurchen tribes be called Manchus, and in 1636, after receiving the submission of the last Khagan of the Mongols, he proclaimed himself to be Emperor of the Qing Dynasty. In 1637, he formally incorporated Chinese people into the Banner System.

The result of these military and political reforms was that Manchu Qing armies were now the equals of the Ming forces, and between 1640 and 1642, the Ming forces in the North-East collapsed, allowing the Qing to control all of the territory that was north of the Great Wall. Hong Taiji did not at first want to conquer China, being content with ruling the northern lands. He offered to accept a title lower than that of emperor and maintain the territorial status quo if the Ming would give him regular payments and other support. Only when the Ming refused to do this did he set out upon the goal of conquering China after receiving advice from Chinese advisors. But he died in September 1643 as his armies advanced against Shanhai Pass, the only major Chinese fortress (within the Great Wall) the protected the Northern Chinese plains from the Qing armies.

The Ming dynasty at this time was riven by corruption and economic disruption which led to massive uprisings. One of these uprisings, led by Zhang Xianzhong who had been in rebellion since 1635, conquered the whole of Sichuan in 1644 and proclaimed the Xi Dynasty. Another uprising, led by the peasant Li Zicheng who had been rebelling since at least 1633, was able to conquer Beijing as well as much other territory in 1644 and the Chongzhen Emperor committed suicide in Beijing. Li proclaimed himself to be founder of the Shun Dynasty. Wu Sangui, the commander of the largest Ming army in Northern China in 1644, decided that rather than try to restore the Ming Dynasty in Northern China, join the Shun Dynasty, or in theory try to become his own emperor, he and his army should join the Qing. With this defection, there was no need for the Qing to besiege or storm Shanhai Pass. Qing forces, augmented by Wu’s army, destroyed the Shun dynasty in 1644-1645, so that the Qing took over northern China. The remnants of the Ming dynasty (the Southern Ming) were too corrupt, demoralized, and weak to resist effectively Qing conquest (although they broke the power of the Xi dynasty enough that the Qing easily conquered Sichuan and killed Zhang Xianzhong), and the last remnants of the Ming were conquered (in Taiwan, as vassals of the Kingdom of Tungning) in 1683.

From this summary, it is clear that infighting and corruption on the part of the Chinese played a major role, but so did Manchu willingness to be multi-ethnic, multilingual, open to adopting new ideas, and willing to learn from mistakes.