I would preferably wish for the reactions of the United Kingdom and Russia, due to both having a vested interest in the affairs of Central Asia.
Thank you in advance!
Before getting into the thick of it, it is worth bringing up that 'Dungan Revolt' is somewhat of a misnomer. It is true that the Qing themselves tended to group the Muslim revolts in northwest China with those in Xinjiang, such as in later commemorative art, but most modern historians prefer to understand the revolts in the two regions as distinct events with distinct causes. This I get into more in this answer. Because your question is specifically about Xinjiang and Yaqub Beg's regime, I'll only be discussing that (not least because I wouldn't even know where to start on foreign perspectives on the revolt in Northwest China!)
The exact extent of Qing power in Central Asia had always been somewhat variable. While control of the Tarim Basin and Zungharia was undisputed, how far Qing power extended beyond the Ili Valley was a little questionable. Officially, the Qing claimed suzerainty over the Kazakhs as far as Lake Balkhash, a claim that was contested rather obviously by the Kazakhs themselves, and also by the Russians. However, an uneasy truce had lain over this region since the Qing conquest of Xinjiang in the 1750s, with the Kazakhs appealing to both sides as far as they could. Things would change by the 1860s.
Russia's advance through Central Asia in the nineteenth century was one with several causes, but the critical one was a perceived need to ensure that Russian subjects and property in northern Central Asia were kept safe from nomadic raiding from the south – not, despite British paranoia, an ambition to conquer British India. In fact, despite the popular British belief that they were involved in a 'Great Game' for control of Central Asia with the Russians, it was a game that only the British were playing, and yet one they seemed to do exceptionally poorly at (cough cough Afghanistan). Russia's aims were by and large regional, and while they did occasionally seek to extend some degree of influence in India, or exploit British paranoia about a Russian invasion for diplomatic capital, any serious invasion of India was off the cards due to logistical constraints alone.
From the late 1840s to the 1870s, the Russians established fortress lines deeper and deeper in the steppe, only to find the logistics of sustaining these forts virtually insoluble. In the long run, the Russians came to conclude that the only viable option was to conquer southern Central Asia outright. This conclusion was not rooted solely in pragmatic assessment as such. The Russians' choice to counter nomadic raiding through warfare rather than diplomacy had been heavily influenced by their conviction that 'Islamic fanaticism' necessarily put relations between them and the Central Asians on an inherently hostile footing – something that would influence their view of Yaqub Beg considerably.
The capture of the Kokandi fortress of Aq Masjid – coincidentally governed by Yaqub Beg – in 1853 served as the first critical stroke in a series of campaigns over the next few decades. When, in 1864, a mass revolt of both Sinophone and Turcophone Muslims in Xinjiang led to its sudden severing from the Qing Empire, the Russians had already been gearing up for strikes deeper into Kokand, with Tashkent, the seat of 'Ali Quli Khan, coming under siege from troops under Mikhail Chernyayev late in the year. It was due to this Russian invasion that a Kokandi force under Yaqub Beg, a loyalist of the deposed Mallā Khan, made a break for Xinjiang and attempted to establish a new base. In the event, this came too late for Tashkent, which fell to the Russians in June, leading to the establishment of a Russian protectorate over Kokand.
Even before Yaqub Beg consolidated control of the Tarim Basin in the wake of his victory at Khan Ariq in the summer of 1865, the Russians viewed the revolt in Xinjiang with great trepidation. While the Qing had been a rival, they were, to quote Alexander Morrison, a 'known and predictable quantity': they had relatively little ambition of expanding further, and sought at most to ensure that their claims were acknowledged. The revolt meant that Xinjiang had come under the control of the very 'Islamic fanatics' of whom the Russians had already been so paranoid, and led to concerns that the broader Central Asian frontier would become dangerously unstable. As such, Russia would eventually cement its hold over the Lake Balkhash region, and by 1871 would even take over the Ili Valley proper – then under the rule of the somewhat obscure rebel leader Abu'l-'Ala Khan – ostensibly for 'safe-keeping' but with no real expectation of a Qing reconquest of the region any time soon.
Britain, conversely, saw the emergence of a new polity in the Tarim Basin as a point of particular opportunity, and pursued a relatively proactive policy in the region in response. Breaking with their earlier policy of 'masterly inactivity', the British government in India attempted to build up Yaqub Beg's state in Kashgar as the linchpin of a wider network of clients and buffer states between the British and Russian empires, and sent a series of envoys and eventually formal ambassadors between 1868 and 1874. This was not merely token support, however: these British missions entailed substantial military aid in the form of several hundred rifles and revolvers of various patterns, and more importantly machining expertise that allowed the Kashgarians to establish an arsenal for producing modern firearms themselves. One Russian visitor in 1875 alleged that several British engineers were working for Yaqub Beg, and that his arsenal was turning out at least 16 new rifles per week, having already managed to convert some 4000 muzzle-loading weapons to breechloaders.
Yet both powers ultimately came to similar conclusions regarding Yaqub Beg's regime, even amid British attempts to establish it as a client state. By the early 1870s both had concluded that it was here to stay and was an essentially legitimate entity, and that as a result some sort of settlement would have to be made with it. Russia, in fact, would be the first to do so in 1872, signing a commercial treaty with Yaqub Beg (identified as the ruler of Yettishahr) which stipulated a flat 2.5% tax rate on goods transported over the border, and also that Russian and Kashgari merchants would be permitted to:
The British treaty two years later in 1874, this time referring to Yaqub Beg as the ruler of Kashgar and Yarkand, was slightly more comprehensive than the Russian treaty in its stipulations, and also largely concerned commercial affairs. As with the Russian treaty it specified the tax rate on cross-border movement of goods (zero in the case of goods moved from Kashgar to India; 2.5% in the other direction) and stipulated the right for merchants to travel and trade in each other's territory; it also covered additional specifics, including but not limited to an exhaustive list of what combination of defendants and plaintiffs led to what jurisdiction a case would be tried under, and what would happen if a British subject died in Kashgari territory.
But while the treaties implied a sort of amicable approach to the Kashgarian regime, the two powers privately did not see it as particularly sophisticated nor reliable. Both still saw him and his subjects as part of a broader continuum of Turkic Muslim 'savages' in Central Asia who would ultimately be subsumed by the 'civilised', Christian empires astride them. H. W. Bellew, the British envoy to Kashgaria in 1873-4, had 'referred to the small khanates of Bukhara and Khoqand, recently conquered by Russia, as "happily doomed" because of their fanaticism and intolerance', to quote I. W. Campbell, and had similar attitudes towards Kashgar. Campbell highlights that despite Britain's effective strategic alliance with Yaqub Beg, and thus 'every incentive to depict Ya’qub Beg as a ruler the British could work with', Bellew could not help but depict him in the mould of a 'savage' ruler whose regime was characterised by 'disorder, oppression, and despotic caprice', much like his Russian counterparts did. And both sides were chiefly interested in the material benefit to be derived from his regime in the form of commercial opportunity, rather than the actual legitimacy of his state or the quality of life it did or did not provide for those living under it. Well, and Britain believed that it needed allies to keep Russia in check (though paradoxically, a Kashgar being constrained as a British puppet would probably have been preferred by the Russians to a fully independent polity.)
Russia's soft opposition to Yaqub Beg intensified in 1875 when the Ferghana Valley, within Russia's new Kokandi client state, erupted in revolt. The brutal suppression that followed, while on some level quite typical of imperial campaigns of its sort, was also partly motivated by the threat of Yaqub Beg's involvement, as the atrocities committed against the rebels and civilian population were at least partly calculated to discourage him from expanding outside of Kashgar and attempting to seize the khanship in Kokand as well. Mikhail 'Bloody Eyes' Skobelev, perhaps the most famous and infamous of Russia's 'Turkestan Generals', made his name – in both senses – starting with this campaign, having been in transit through Ferghana on an embassy to Yaqub Beg that had been aimed to settle the Kashgar-Kokand border; despite the cancellation of the embassy, Skobelev still achieved this end when he drew the border in blood.