I am unfamiliar with the fate of Constantinople after the 1453 Ottoman conquests but I can speak a bit about the city in the wake of the Fourth Crusade. The short answer is that, yes, we have accounts of the various treasures that the Byzantines and French crusaders took from Constantinople after the sack of 1204. Some of these treasures are on display in Venice like the Tetrarchy and the Horses of Saint Mark while others (specifically holy reliquaries) made their way across Europe. Prior to the Fourth Crusade, Constantinople had been a hub for the remains of saints and crusaders were intent on taking back whatever relics they could to their homes. I highly recommend the book Sacred Plunder: Venice and the Aftermath of the Fourth Crusade by David Perry (Penn State University Press, 2015) for an in-depth examination of the fate of these relics. Perry makes substantial use of a genre of text called “Translatio Narratives” that consider the transmission of relics from the Holy Land to Europe (the third chapter of the book explicitly deals with this). You can check out sections of this monograph through Google Books here.