I remember reading about the Roman and Byzantine Empires as a kid and thinking it was kind of odd that Rome was ever supplanted at all. As an adult I've slowly come to realize that there's a lot of questionable Eurocentrism baked into that attitude, and that the Bosporus was a lot closer to the center of economic and political gravity by this point than the Tiber. That made me realize I'd never really read how Roman citizens and elites viewed this shift from West to East. What did they think?
Let me close by invoking Cunningham's Law thus:
Roman citizens and elites thought the shift of political power from Rome to Constantinople clearly justified and almost obvious, at least if you take as given the feasibility of founding a major new city on the Bosporus. Egypt and Persia were more central to the late Roman world than were Iberia, Britain, northern Gaul, much less the barbaric Germanic lands beyond the Rhine and the Danube. Rome was no longer optimally situated to serve as the main governmental, military, and commercial hub for its namesake empire. Yes, many Roman partisans were disgruntled, but even they could not come up with good arguments against Constantinople's ascension. Only grumpy ones.
Rome was not that important of a city by the time of Constantinople's founding. Since the Crisis of the Third Century emperors were making their capitals near the frontier, like Mainz, Trier, Adrianople, or Antioch. Rome, which represented the power of the Senate and old aristocracy, was old news in a world dominated by the army officer class from the provinces. As the emperors accrued more despotic powers to themselves, fully realized with Diocletian's reforms, influence and power in the empire was based more on access to the emperor than anything else. Rome had symbolic power, but the courts had been absent for a long time when Constantine came to power. The new court at Constantinople, though it was close to both wealth and the frontier, changed nothing for most people who even knew the world beyond their village, except for the emperors who found a convenient place of both luxury and proximity to trouble spots. It may seem backwards to put the capital near the frontier, but the Crisis of the Third Century had proved that wherever the emperor wasn't, invasion and insurrection was.
However Constantinople evolved because of its permanence as the capital. The capital was always moving between frontier cities before, but a settled court allowed new people to seek it as the obvious center of wealth and culture; A real New Rome. Aristocrats of the eastern provinces stayed there part-time to curry favor with the emperor and important bureaucrats, while the poor went seeking new opportunity.