In 1204, as I noted in The Elgin Marbles, the Fourth Crusade went horribly wrong and the Crusaders took Constantinople. The relics that the Byzantine Emperors had not already sold off to raise funds, were soon shipped to the West, and in their wake there followed antiquities. The Porphyry Tetrarchs went to Venice, built into St Mark's Basilica, with only a foot left behind in the Bodrum Camii. As did the Classical or Hellenistic horses probably one of the two such groups from the Hippodrome, and two pillars from St Polyeuktos which also made their way to San Marco ...
There may have been some antiquities in Byzantium, but most were brought by Constantine when he refounded the city as the new Rome. Justinian is the last emperor known to have brought antiquities to the city: two horses from the Artemision at Ephesus to the Chalke Gate (Patria II, 75).
In the De Signis (647-55) Nicetas Choniates describes the sacking of the Hippodrome, with statues that had survived since the age of Pericles destroyed or stolen. A giant bronze of Heracles by Lysippus stood in Tarentum until 209 BC, then moved to the Capitol in Rome, and finally to Constantinople under Julian ... it was knocked over during the sack (649-50), was lying on the ground in 1561, but is now lost.
Statues of Julius Caesar, Augustus, Diocletian, Theophanes of Mytilene, countless Byzantine Emperors all vanished, possibly into European collections. Countless sculptures that the Romans had taken as spolia, now became trophies taken home by the Crusaders.
What's interesting given the controversies over the dating of the She-Wolf in the Capitoline, is that Choniates (650) mentions a similar sculpture in the Hippodrome, which he believes was melted down (although he wrote from exile, so may not be correct):
Nor indeed did they keep their hands of the hyena and the she-wolf which Romulus and Remus sucked. For a few staters, and what is more, copper, they consigned these ancient and revered objects of the nation to the smelting furnace.This cannot be the statue attested outside the Lateran Palace in Rome from the 10th century on, but might be the famous group mentioned by Livy, Pliny and Cicero, once in the Capitol, since the Lupercalia festival was celebrate in the Hippodrome in Constantinople until quite late. Just as the Lupercal celebrated the foundation of Roma, so the festival celebrated the foundation of Constantinople as the New Rome - and this makes it more likely that the group in Constantinople was brought from Rome to reinforce this point.
This Late Antique mosaic from Piazza Armerina shows the Circus Maximus in Rome, and gives us a good idea of what the Hippodrome in Constantinople would have looked like, decorated with statues:
The statues in the Capitol and the ones in the Circus Maximus are different groups, and show how important the motif was to the Romans.
The Chronicon Paschale is clear that the hippodrome in Constantinople deliberately imitated the Circus Maximus, so it's not surprising to find the She-Wolf in both (Ol. 277.1), since there were Egyptian obelisks in both. Although we think of the Roman Circus as being the location for chariot races, it was also the venue for ceremonies such as the 'Triumph' of Belissarius.
Although there were many circuses around the Roman world, the Hippodrome in Constantinople and the Circus Maximus in Rome were unique for having obelisks, a Lupercalia, and key sculptures - which is why their loss is so sad.