You pose a very interesting question. Constantinople was indeed heavily affected demographically due to the loss of Egypt, which was indeed still the breadbasket of the Roman world. However before I go into this and answer your question directly I feel it's important to give some demographic context in the centuries prior.

Constantinople reached it's demographic peak during the early reign of Justinian (c. 530 A.D), estimates from this time range from 500,000 to 1 million people. Having been unaffected by the turmoil of the 5th Century and given a constant supply of grain from Egypt (as used to be the case with Rome) it was without a doubt the largest and most impressive city in the west, if not the world at the time.

This came to a screeching halt in the 540's however with the onset of the Plague of Justinian. When the plague had finally died down about 40% of Constantinople's population had been killed by it. However the city still had a prospect of recovering, thanks to the free grain it could encourage migration to the city and sustain a large population, aswell as economic growth. However the late 6th and 7th Century was not a good one for Imperial stability and as such Constantinople's population hovered possibly around 300,000.

The prospect for demographic recovery however saw a sharp turn as the wars with the Sassanids got worse. When Heraclius usurped the Imperial throne from Phocas, the new Emperor supposedly considered moving the capital to his previous home of Carthage. Whether or not this consideration was a charade or not is not really important, because the mere thought of the Emperor abandoning the capital was enough to convince the elites of the city to temporarily end the free grain from Egypt, which had fallen to the Sassanids in 618.

This event should not be overlooked when discussing Constantinopolitan demographics, as it meant the city could no longer sustain the massive population it previously had. The population must have then presumably fallen substantially, if not from the cancellation of free grain, then the attempted Avar/Sassanid siege of 626 must have convinced many people to leave.

The Romans would eventually pull out a pyrrhic victory from the war with the Sassanians, and as you probably know would retake Egypt for a few years. When the Arabs bursted out of the Arabian Penninsula and took Egypt from the Empire, the prospect of restoring the free grain policy was dead, permanently.

When the Arabs captured Alexandria they gave the Greek elites of the city a choice. They could stay and loose their wealth, status and property, or leave for Constantinople with the wealth they could carry. Thus throughout the 7th Century Constantinople would become a shelter for refugees from formerly Roman provinces.

One of these refugees is actually quite noteable, that being Kallinikos. Kallinikos was a Christian refugee from Syria traditionally credited as the inventor of Greek Fire proper, which helped turn the tide against the Arabs in their two attempts to conquer Constantinople itself.

By the turn of the 8th Century Constantinople would have been a sorry sight compared to the early 6th Century. It's population may have been as low as 200,000, it's good and services would have been devestated by years of war, plague and siege. Many buildings would be abandoned or ruined. However Constantinople, unlike many other cities in the Empire like Nicomedia or Carthage managed to retain urban continuity due to it's strategic position.