I suspect that the odds are very strongly against any polity surviving a very long time. It depends on what counts as still being "the same" of course--was Qing Dynasty China the same country, the same empire in fact, as Han China? Was the Tang Dynasty the same as Han? Arguably if they are all just different dynasties of one Empire than one might say modern Communist China is also the same old Chinese Empire!
And one might say the nations of Europe are still in some sense or other Rome.
But the latter two statements seem sillier than to suggest that the Ming Dynasty was continuous with the Han, and it seems especially preposterous to claim the Roman Empire survived.
Clearly there is a lot of turbulence in human affairs. I think the bigger the Empire, the greater the chances of things going really wrong on a continental scale, and the failure of the Empire to hold on to some territories, no matter how peripheral, will lead to cascading failures of confidence and boiling divisiveness that must eventually cause a final and dramatic collapse.
"The slow one now will later be fast"--as the world situation changes, as economies evolve and various peoples evolve new societies, political strategies and forms that worked well in one time and place won't work so well later and elsewhere, aside from the probability that success breeds its own failure by ossifying institutions that, whether or not they are in theory flexible and capable enough to manage new situations, are now in the hands of people of a different mentality than those who founded them.
Insofar as China is the same China as there was in Roman times, it is because the Chinese evolved ways of incorporating revolution itself into their system. Similarly India has an identity spanning thousands of years because its society largely divorced itself from identifying with particular political regimes. But mostly, it has an identity because it is a distinct, sharply bounded region geographically, in which certain cultural patterns circulated freely but were to some degree kept semi-isolated from others, so along came various conquerors, culminating with the British, who as outsiders conceived of the place as one nation and forcibly unified it. Barring a really massive decimation of human population I have every confidence that thousands of years hence both China and India will be quite distinct nations, but their political systems may have no continuity with today's whatsoever.
So--if some particular imperial core does persist, and keep expanding, and never fails in many thousands of years, either we are looking at a very improbable timeline (meaning it holds few lessons for us) or we are looking at a society that has somehow evolved some secret to persevering and resisting both internal decay and exterior attack. If these methods involve some sort of oppressive authoritarianism, I'd think sooner or later the malcontents would prevail and overthrow such a regime.
So I think you must be talking about some people who manage to learn deep and valuable lessons about politics and society, who have governance down to a science and widely cultivated art. Such a successful government would have to be flexible, clever, good at self-criticism and self-renewal; very likely syncretic, able to recruit followers from all manner of foreign societies.
It would probably be rather exciting and beautiful. It would also be very very hard for us to imagine such a dynamic Utopia!
I'm not saying it could never exist, but if we had any clue how, we'd be living in that sort of society and not the blinder ones we do live in.
If we write off such a subtly grand empire as ASB, then we are left with the inevitability of failure of the big players of one generation eventually; with the constant ferment of new societies emerging, new nations rising to prominence often from very humble beginnings, entire new international economic systems rising up behind everyone's back--all manner of factors will keep the Wheel Of Fortune spinning.
I'm game enough for trying to envision an Empire of Wisdom and Grace that can foresee these trends and navigate them, cultivating here, diverting there. But I doubt we'll come up with a plausible one!