In terms of public officials, the Senate of Constantinople (or Byzantine Senate) was intended both for Roman officials who already lived there and to encourage more officials to make the journey east; Constantine would also offer free land and grain as an incentive for these potential newcomers.

The Senate of Constantinople was continuously and increasingly handicapped, working in dialogue with the Council to provide legislature at the emperor's behest and as a bit-part consultative board and judicial court when needed. It's lack of prominence in the city was inadvertently confirmed by an 8th Century Byzantine writer in their guide to Constantinople (Parastaseis syntomoi chronikai) which, when describing the building, hazarded a hopeful guess that it must be named after some guy named 'Senatus'... Nevertheless, the Senate of Constantinople was clearly intended as a continuation of the tradition and prestige of Rome, fitted with senators descended from its former glory.

As for the Western Roman people in general, there was little incentive to move such a considerable distance; many of the wealthy families had acquired a considerable amount of land in the West, often evading a broken taxation system with ease - they would have seen little reason for long-distance emigration.

Some trades may have seen increased connections with the East, perhaps following the money along the new trade routes. But, once the new capital had been settled and Constantine's incentives had concluded, emigration would have slowed considerably. Rome and Constantinople would have continued along increasingly divergent paths, lessening contact, the one interconnected empire soon following two disparate destinies.