This question is one that was perhaps more complicated than you may have imagined while writing it, because it gets into two main areas: matters of ethnic reification and classification, and the 'dynastic cycle'. Because the terms 'Hui', 'Tibetan', and 'dynasty' are less concrete, in a grand historical context, than you might imagine.

Problems of Ethnicity in Pre-Modern China and Inner Asia

You allude to the idea of there being 'five main races' in China, but this is something with a particular historical context. The idea of 'Five Races Under One Union', encompassing the Han, Manchus, Mongols, Tibetans, and Muslims, was first promulgated around the time of the 1911 Revolution as a means of reconciling nationalist demands for Han majority rule with imperial ambitions of holding on to the Inner Asian regions of the Qing empire. In practical terms, it was short-lived as a political doctrine, as Mongolia and Tibet declared and achieved independence that year and would not be absorbed into the Republic of China (although it did exercise claims to these regions and technically still does). The People's Republic of China, established by the Communists in 1949, defines 56 ethnicities: the Han, and 55 state-designated 'minority ethnicities'. Neither of these classifications has any objective correctness, and both are ultimately Procrustean to some extent or another. The Republican 'Five Races' notably exclude any of the indigenous peoples of southern China, while the Communist method of ethnic classification arbitrarily combines several groups that self-identify in distinct ways, in some cases purely on the basis of a shared character in their Chinese transcriptions rather than any real consideration for indigenous genealogies.

But in turn, these classifications also reflect the scope of a larger 'China' as created by the Qing, whose conquests beyond China Proper established a (mostly) lasting political settlement over large portions of Inner Asia that had not previously been unified under a single state formation. I use 'China' in scare quotes because there was not, and arguably still is not, a single term in the Chinese languages that corresponds to the relatively vague English use of the term. 'China' can be defined as a geographical space, a cultural space, or as a political entity. The Republic of China and People's Republic of China have long had a certain interest in enabling that vagary in order to justify their existence as being nation-states, rather than as empires, claiming large parts of Inner Asia that had been outside the scope of historic Chinese empires into the geographic and cultural space of 'China', on the retroactive basis of their current political scope. Take for example Tibet. Historically, Tibet was never geographically or administratively part of 'China' as such before the Communist conquest and annexation in 1951 – while it had been part of the Mongol Yuan and Manchu Qing empires, in neither case was its administrative structure subordinated to or brought in line with the Chinese provinces.

That is not to say that Tibetans or Muslims did not have a significant place in the Qing Empire. Whether the Republicans realised it or not, they were in essence attempting to codify an informal understanding that had developed in the Qing imperial court in the late 18th century, that being what Pamela Crossley terms 'constituencies' and what James Millward calls 'blocs' – Manchu, Han, Mongol, Tibetan, and Muslim. The issue is actually quite complex and theory-heavy – Crossley's 'constituencies' are ideological concepts and not actual people, whereas Millward's 'ethnic or linguistic blocs' (reframed by Emma Teng as 'culture blocs') are actual groups of people, at least as construed by the court – but we need not dig too deep into the weeds here. The main thing is that the idea of Tibetans and Muslims being core ethnic groups in China is and was a product of the eighteenth-century Qing conquests, rather than being rooted in any prior imperial Chinese rhetoric. Indeed, we can extend that to the Manchus and Mongols, whose homelands lay outside the scope of any Han Chinese empires. The Qing conquests enabled the prospect of conceptualising – or at least rhetorically claiming – 'China' as a multiethnic union.

Before moving on it is also important to raise the matter of Muslim identity as classified by imperial and republican Chinese states, because 'Muslim' in the case of the 'Five Races' does not map neatly onto the Hui, a category which itself has a complicated history. Under the Ming, all Muslims were referred to as 'Huihui', a corruption of the earlier 'Huihe', a transcription of 'Uighur'. Under the Qing, there was some effort to distinguish between subgroups – the main groups being the Sala Hui in the Kokonor region, Chantou (Turbaned) Hui in Qing Turkestan (now Xinjiang), and Han Hui for Chinese-speaking Muslims, located primarily in Gansu, Shaanxi, and Yunnan. The broad-brush use of 'Hui' to designate all Muslims in the empire was retained in 1911, but fell out of use during the Republican period as Muslim identities in Inner Asia, especially that of the Uighurs, solidified, leading to 'Hui' becoming a term specifically for Sinophone Muslims in China Proper by the time of the PRC's founding, and becoming reified as such by its minority policy. In effect, then, 'the Hui' as an ethnic group, as understood today (that is, Sinophone Muslims), did not exist during the imperial period.

Problems of the 'Dynastic Cycle'

The periodisation of Chinese history as a series of 'dynasties' is, for many reasons, problematic, as I discuss here. One of the major problems with a dynastic-centric periodisation is a paradoxical effect on how change and continuity are perceived: there is often a presumption that a 'dynastic transition' implies substantial political changes, but that in the broad sweep there is little effect on the underlying state structures of 'China'. But another issue, one much more relevant to your query, is that there is a sort of 'there can be only one' problem when it comes to the imperial claim: the history of imperial China gets presented as a bit of a relay race where the torch is passed from one singular 'dynasty' to the next at discrete points in time, with the occasional mess of squabbling states such as the Northern and Southern and the Five Dynasties and Ten Kingdoms periods. The problem is that these states did not conceive of themselves as simply inheriting each other's mantles; nor did they see themselves as coming into existence at the moment that their 'predecessor's' existence ceased. To cite just one case, the Qing were founded in 1636, nearly a decade before the fall of the Ming proper in 1644, and even then remnants of the Ming court and its loyalists continued to operate as the 'Southern Ming' until (nominally) 1683, while several rebel states, most prominently the Great Shun, also made claims on the imperial title. Starting the Qing in 1644 is an arbitrary, Sinocentric approach to the issue.

The most substantial elision resulting from this 'relay race' model is that the period between the end of the Five Dynasties and Ten Kingdoms in 960 and the establishment of the Yuan in 1271 often gets referred to as 'the Song', despite the fact that the Song themselves never actually controlled all of China proper: to the northeast, the Khitan-established Great Liao held several areas of historically Han settlement, including the Liaodong region and what is now northern Hebei (the region around Beijing); they would be usurped by the Jurchen-established Jin, which went further by conquering all of China north of the Huai river and driving the Song to the south; to the northwest, after a brief period of Song dominion the Tanguts revolted in 984 and eventually established the state of Western Xia in Gansu in 1038. Western Xia was, for all intents and purposes, the weakest of the three states, and would be the first to fall to the Mongols, but it nevertheless exercised a claim to imperial status. In the traditional chronology, and even in the modern trend for referring to 960-1271 as the 'Song-Liao-Jin' period, Western Xia is rather unceremoniously cast aside.

Summaries and Conclusions

I bring up Western Xia because in a sense, it is a decent candidate for being a 'Tibetan dynasty' as phrased in the question, but there is a reason I say a decent candidate and not a definite contender. The Tanguts had a relatively long history of entanglement with Tibet, but from a linguistic standpoint they were relatively distant, with Tangut being part of the Eastern rather than the Western Tibeto-Burman family and thus arguably closer to modern Burmese than modern standard Tibetan. More importantly, 'Tibetan' as an identity was not universally accepted by those living in what we would now call Tibet, as the territorial extent of Tibet waxed and waned considerably over the course of its history, with perhaps the most substantial waxing taking place during Qing rule/patronage/whatever you want to call it. In effect, we run into the problem from the first section again: ethnic identities cannot simply be retrojected back through time as though they have existed forever, let alone been unchanged forever.

So, have there ever been Hui or Tibetan dynasties of China? No, firstly because 'Hui' and 'Tibetan' as identities are more recent than they may first appear, and secondly because Tibetans in particular were not a substantial part of 'China' in any meaningful sense until arguably the Yuan and more concretely the Qing (and even if you count the Yuan they were not part of the Ming). Moreover, the 'dynastic' classification breaks down once you get in the weeds a bit, and is just an unhelpful method of categorisation except in the broadest strokes.