The term 'Panthay Rebellion' is a useful shorthand, but also a somewhat misleading one. Firstly, 'Panthay' is very much an exonym, originating from British colonial authorities based in Burma, rather than a self-appellation. For the most part, Muslims in the southwestern regions called themselves Hui 回, a distant corruption of 'Uyghur'. That's not to say that they were all actually of Turkic origin, indeed many, particularly in the northwestern provinces of Gansu and Shaanxi, were descended from Han Chinese converts. Secondly, there was not a singular, coherent rebellion. David Atwill, in his study of the uprisings, discerns three separate rebel tendencies, roughly corresponding to Yunnan's three major geographical divisions: a western portion centred on Dali, an eastern portion centred on Yunnan-fu (now known as Kunming), and a southern portion centred on Pu'er (of tea fame) and Lin'an. They by no means acted in concert, although their original motives were indeed broadly aligned. Thirdly, it was not only Muslims, but indeed a large portion of the province's population which included Han Chinese and, crucially, aboriginals, that rebelled.

Let's start from the second part of your question: why were there Muslims in Yunnan? After Taiwan (absorbed in the 1680s) Yunnan was the newest region to be incorporated into the territory of China, being made a province during the early Ming Dynasty in the early 1380s. Before that, it had been conquered by the Mongols in 1253, making them the first China-ruling dynasty to exercise control over the region. Muslims were considered loyal administrators by the Mongols (even though the Yuan were the only one of the four major khanates not to eventually convert to Islam), and so many were moved into Yunnan to help maintain control, especially as the region's population consisted predominantly of various aboriginal peoples like the Miao, Tai and Zhuang. Some Han also filtered in over the years, but the Muslim population was the one specifically introduced by the Mongols. After the fall of the Mongol empire in China and the conquest of Yunnan by the Ming, the Muslims remained and retained a degree of prestige under the Ming government, with the most prominent of these being a certain Zheng He, who was a direct descendant of one of Yunnan's Muslim governors under the Mongols. By 1860, it was estimated that Yunnan's population was around 50-60% aboriginal (termed yi 夷 in Chinese-language documents), 10-20% Hui, and 30-40% Han Chinese. While cities contained a mix of peoples, villages were usually mono-ethnic in terms of makeup, with distinct Han, Hui and aboriginal settlements largely erratically scattered across the province.

However, this figure obscures certain demographic trends. Between 1775 and 1850 there was a major surge in Yunnan's population, which leapt from an estimated 4 million by the earlier date to 10 million by the latter. While this must be considered alongside a general surge in China's population in the early modern period, which went from perhaps 150 million in 1650 to 450 million in 1850, a large part of the province's population growth came from the migration of Han Chinese settlers to the region, some being agricultural colonists but many seeking work in the region's extensive mines, which produced a large portion of the empire's silver and copper. It was in part this migration, Atwill argues, that eventually prompted the uprisings of 1856-73.

Before I go into that, it's important to explain that there is somewhat of a controversy, still somewhat unresolved, as regards these categories of Han, Hui and aboriginals in the period in question. Just as opinions have been divided over whether the Manchus possessed an 'ethnic' identity before the state-building initiatives of the mid-18th century, the question of whether the Hui saw themselves as possessing an ethnic identity, a religious identity, or indeed any sort of clear identifying framework, is also somewhat unresolved. Atwill suggests that there was a Hui ethnic identity independent of Islam, but Jonathan Lipman, in his study of the Gansu and Shaanxi Muslims, argues the opposite: that Islam was a unifying influence beyond ethnic differences. To explain, Atwill draws on writings by Muslim elders in Yunnan that suggest that the Han were corrupting their youth away from Hui teachings, but that this would make them Hanjiao 漢教 – 'Han-taught' – without necessarily ceasing to be Hui; also that Yunnanese Hui were not exclusively Sinophone, but that many also adopted aboriginal languages if they happened to be more proximate such aboriginal groups, yet retained common identification of Hui. Lipman, on the other hand, suggests that there were ethnic differences perceived between the Sinophone Hui and Turkophone Salars, but that they nevertheless showed such solidarity that this went unnoticed, and both Muslims and Qing officials saw the main point of divergence as being sectarian adherence. Bear in mind, then, that because I'm drawing mainly on Atwill, I'm continuing the view here that ethnicity rather than religion was the key matter at hand, but that there is certainly a framework in which you could view it otherwise, even if it has not (yet) been applied to the Yunnan context.

Irrespective of their differing assessments of religion versus ethnicity in terms of identity construction, the two sides do agree on one thing: that the Qing's policy in these regions with significant Muslim populations was one marked by a significant anti-Hui bias. In particular, Hui were seen as inherently more violent than Han, and invariably Han were favoured in legal cases involving both religious groups. Was this warranted? Of course not – it was a complete stereotype. However, it didn't come from thin air. Hui soldiers had a strong reputation for valour (indeed, Hui troops from Gansu would form one of the most comparatively distinguished units of the regular forces that supported the Boxers in 1900), and were disproportionately represented in the military officers' examinations (perhaps because unlike the civil examinations, which were much more explicitly based on the core texts of Confucianism, they were less religiously objectionable).

Nevertheless, the bias was real, and this is perhaps best borne out in the response to the Baoshan Massacre of October 1845. The Baoshan Massacre had been the product of over a decade of growing tensions resulting from the formation of Han 'Incense Brotherhoods', a cover for what were essentially roving gangs of anti-Hui bandits. When Hui villagers in the Baoshan area began gathering in self-defence, the brotherhoods vandalised a mosque, leading to reprisals from Hui bandits solicited by the villagers in defence. These gradually gathered into a rebel group under Ma Da numbering several hundred and conspired with elements inside the city. Pre-emptively, the brotherhoods and affiliated Han militias, with tacit government support, marched into the city and massacred most of its Hui population, as many as eight thousand according to some reports. Qing officials asserted that the entire matter was officially sanctioned as a move in self-defence, but He Changling, the viceroy of Yunnan and Guizhou, changed his tune somewhat between reports, already cause for suspicion, before four Hui leaders, including one Du Wenxiu, travelled to Beijing to petition the emperor for redress. In response, the Qing appointed a new governor-general, a certain Lin Zexu ((in)famous for his role in the run-up to and early months of the Opium War with Britain).

Lin laid the blame on the brotherhoods for the massacre, and accounted for the role of Han Chinese in perpetuating rumours of Hui attacks, but still made two major omissions or errors. Firstly, he claimed 4,000 Hui deaths rather than 8,000, and secondly, he ignored the acquiescence of the city's officials to the carrying out of the massacre, claiming simply that they were overwhelmed. He did manage to capture over 300 brotherhood members in a first round of arrests, of which nine Incense Society leaders were executed and a further 137 received corporal punishment, but in his broader actions Lin nevertheless revealed a disturbing pattern of anti-Hui inclination. He claimed that in Yaozhou department, the Han saw eight times as many casualties and nine times as much property damage, and of 138 executed prisoners, 101 were Hui. Moreover, he liberally applied harsher statutes applying exclusively to Hui with regards to punishments, such as one that made any group of three or more Hui involved in criminal activity, where any of those Hui were armed, punishable by compelled military service. Furthermore, while he recognised the role of land disputes as a root cause of violence, his solution was decidedly insensitive, as he had many of the Hui in the Baoshan area relocated into a zone along the banks of the Salween River, infamous for being one of the province's most malaria-infested regions.

On the matter of these disputes, it is worth bringing up now that Atwill discerns a distinctive trend in how Hui-Han relations actually operated. While a decisive majority of the Han were post-1775 immigrants, those from older lineages actually appear to have taken the Hui side in many of the disputes involved. 'Older' Han had developed a much more accommodating relationship with the Hui and aboriginals, to the point where some may even have considered themselves Yunnanese first and Chinese second, in contrast with 'newer' Han, who were keen on displacing the local populations, or even, as evidenced by the brotherhoods and the Baoshan massacre, outright exterminating them.

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