So the answer partly has to do with the religious historiography of Islam, partly to do with actual historical developments, and obviously quite a lot to do with convention.

Firstly, the Rashidun, Abu Bakr, Umar, Uthman, and Ali, were not closely related and did not succeed one another as a coherent dynasty.

From the Umayyad perspective however, there is a significant degree of overlap. According to the Umayyads, Uthman would have been the first Umayyad caliph.

Traditionally (i.e. since the 8th/9th century and the rise of the Abassids) this view is rejected. The traditional religious view is that the Rashidun (which means "rightly guided") exercised their role as God's agent and the successor of the prophet as righteous Caliphs. The Ummayads in this conception were merely earthly kings, and were a corrupt regime at that. So in the traditional religious sense you have a very real transition of state.

For historians the real problem is that there is much debate, prior to the reign of Abd al-Malik towards the end of the 7th century, as to whether the Islamic Empire was a state at all. There's debate over terms like "Islamic Empire" because there is very little that is obviously Islamic about it. The caliphs prior to Abd al-Malik didn't make Arabic primary, didn't leave Islamic inscriptions anywhere, didn't build any major buildings, didn't really alter the coinage. Some like Fred Donner posit that much of this must have happened prior to Abd al-Malik but probably not much earlier than Muawiyah, under whom you see early versions of some of these institutions of state develop (for instance, Muawaiyah built a dam in southern Arabia, and we have some coinage calling him commander of the faithful in Persian, but not Arabic.)

For the Abbasids there is equal debate, but probably less so. The standard term for their takeover is the Abbasid revolution, and it involved not just dynastic change but some degree of ethnic and religious claims as well. There's a whole mess of stuff written about this, and while I'm skeptical that it was a proto-Iranian revolution as some authors have written, it nonetheless featured converts and Persians prominently in the makeup of the Abbasid military, had some degree of a claim to an Alid legitimacy, and had other religious elements including the use of the black flag of the prophet.

Moreover, some of the early Abbasids, notably Harun al-Rashid, undertake a level of state building on par with Abd al-Malik that completely reform the structure and form of the empire.

Nonetheless, and despite everything that I've just written, it's still extremely common to see the Rashidun, the Umayyads and the Abbasids listed as the leadership of a contiguous "Islamic Empire," at least until this starts to break up in the Abbasid era as local dynasties take control and the caliph loses much, and subsequently all, of his military and legislative authority.

edit: wrote something that was unclear.