One may just as easily ask how the Qing managed to project power into China, having also had obvious cultural differences and substantial natural barriers between southern Manchuria and China proper. But that's not the question we have here.
The first key point is to suggest that we do not talk of 'Qing China' as such. Not to say that the Qing was not 'China' i one sense or another, but at the same time 'China' is a non-Chinese term that is used to refer to a variety of quite different states, and we ought not to assign labels that imply undue continuity between the Qing and either their Ming predecessors or ROC successors. In practice, the Qing were far more willing and far more able to balance the ideological and practical pressures of exercising rule in both Inner Asia and China proper. Yet even then, it is worth noting that Qing rule in parts of Inner Asia could in fact be quite uncertain even at the empire's height.
While we must obviously note that the Qing Empire was a large system, I think the most productive route would be to first look at the distinct contexts individually, as the Qing tended to approach their empire in a similarly particularistic manner.
It is in many ways ambiguous when we can declare Qing rule to have begun in Tibet, or indeed how far Tibet ever was 'ruled' as such by the Qing. In 1642, contemporaneous with the imminent collapse of the Ming, the Tibetan state of Tsangpa was overrun by the Khoshut Khanate, one of the constituent elements of the fragmenting western Mongolian federation known as the Four Oyirad. However, the Khoshut ruler Güshi Khan then retreated from Tibet and established himself in the region of Kokonor, while in Tibet he set up a 'secular' government known as the Ganden Podrang as well as elevating the Gelug (or Yellow Hat) sect to primacy. From here on out, the relationship between the Khoshuts and the Tibetan state was a sort of patron-protector arrangement, with the Khoshuts providing hard power to the Tibetans if needed, and the Tibetan clergy giving religious legitimation and authority to the Khoshuts.
A similar but more long-distance relationship characterised relations between Tibet and the Zunghars, another Oyirad fragment which had managed to reunite many of the former Oyirad tribes from their base in what is now northern Xinjiang. Galdan, a former student of the Fifth Dalai Lama and the chieftain of the Zunghars, was proclaimed 'Boshogtu Khan' in 1678, a particularly transgressive move as it signalled that the Tibetan clerical establishment now claimed the right to unilaterally confer the title of Khan, previously reserved only for those who were patrilineally descended from Genghis Khan – which Galdan was not. Galdan, in his turn, sought to support the interests of the Gelug sect by waging war in eastern Mongolia, where rulers looked not to Lhasa for religious authority, but to regional and local monasteries patronised by major nomadic leaders. Galdan's wars against the Khalkha in 1688, conducted towards this end, drove them into the hands of the Qing, who counter-attacked and defeated him in battle in 1690 and 1696, soon followed by his death (either of smallpox or poisoning by a lieutenant) in 1697. His successor, Tsewang Rabdan, failed to receive the same sort of reciprocal support with Tibet that his uncle had, and invaded Tibet in 1718, destroying Khoshut power and temporarily establishing control over Tibet proper. This prompted a Qing counterattack in 1720 that successfully drove out the Zunghars and restored power to the Ganden Podrang and the Gelug sect.
How exactly one ought to characterise Qing rule in Tibet after 1720 is a bit of an open question. Yes, it was part of the Qing empire in at least a nominal sense, and the Qing did have a permanently-stationed Manchu official (amban) in Lhasa with the authority to call for military support (which would be employed in 1727 and 1751) and to represent the Qing court, but said official held no formal power in the Tibetan administration. The Ganden Podrang, although theoretically curtailed, remained practically autonomous, and meanwhile the Qing did little about the internal governance of the Tibetan clergy. Tibet's status was thus not so much an official territory of the empire so much as a protectorate state, with effective domestic autonomy in its 'secular' government albeit with the lingering threat of Qing military force, and with basically total latitude in terms of its vast, yet extremely organised and centralised religious hierarchy.
On the flip-side, Qing rulers, particularly the Qianlong Emperor (r. 1735-96/9), demonstrated legitimacy in the Vajrayana Buddhist world through their patronage of Tibetan monasteries, particularly in their key sacred capitals of Chengde and Mukden, and to a lesser extent Beijing. Qing trust in Tibet reached the point where, after the revolt of their Mongol client ruler Chingünjav in 1757, the Qing decreed that future incarnations of the Jebzongdanba khutukhtu, the chief religious authority in eastern Mongolia (whose most recent incarnation was, coincidentally, Chingünjav's brother and who supported his revolt), would only be discovered in Tibet. In addition, the Qianlong Emperor launched two major military campaigns against the indigenous populations of Jinchuan, Tibetan-speaking minorities who had refused to accept Gelug supremacy and had held firm to either the 'Red Hat' sects or even to the pre-Buddhist Tibetan religion of Bön. The second of these campaigns in 1771-6 was particularly notable for the involvement of the lama Rolpay Dorje, who in Qing accounts served simply as an advisor to the Qianlong emperor at Beijing and Chengde, but who in Tibetan accounts was personally present to conjure up dust storms and fireballs in battle – in other words, the Tibetans saw themselves as eager participants in these projects of Qing imperial expansion. Even after a number of strains on the Qing-Tibetan relationship in the late eighteenth century (see below), the Qing saw fit to recognise an expansion of the jurisdiction of the Gelug clergy to include Kokonor and Gansu in the late years of the Qianlong reign. Ultimately, the Qing managed to cultivate what was, on the whole, a very similar relationship to what the Khoshuts had with Tibet, just on a larger scale: the centralisation of religious authority in Lhasa was a desirable end in itself for the Gelug sect, while benefitting the Qing by reducing the autonomy of Mongolian tribal leaders and legitimising the expansion of direct control over the mountain borderland between Sichuan and Kham. Secular power advanced the interests of the religious authorities, while religious power advanced the interests of the secular authorities.
Nevertheless, the extent of Qing control was always tenuous, a fact of which the Qing were well aware. This reached its climax in 1788, when King Rana Bahadur Shah of Nepal dispatched a Gurkha army into Tibet. This invasion proved a limited success as the Tibetans paid off the Gurkhas without the main armies having ever come to blows, but a second and far more successful attack took place in 1791, in which the Gurkhas captured the Tibetans' second-largest city of Shigatse and sacked the Panchen Lama's palace at Trashi Lhünpo, carrying off its accumulated treasure including many donations by the Qianlong Emperor himself. A vigorous counter-attack in 1792 led to a peace deal with Nepal, but the war had served to affirm the fears of Qing authorities who had seen Tibet's autonomy as untenable, most strikingly due to the awareness that eight of the nine senior monks at Trashi Lhünpo had fled before the Nepalese attack that seized the monastery, leading to suspicions of foul play and collaboration – as well, paradoxically, as the execution of the one monk who stayed behind. In the wake of the Nepalese invasion, the Qing implemented the Golden Urn as a means of randomising the selection of new incarnations of lamas, especially but not exclusively the Dalai and Panchen lamas, from a curated shortlist, rather than prior practices in which a variety of methods were used that tended to favour Tibetan elite lineages. Ironically, the Qing would go on to accept many such elites on their shortlists anyway.