Cyril Mango’s excellent article on the fate of ancient statues in Byzantium tells us:
The importation of statues into Constantinople greatly diminished, but did not entirely cease, after the reign of Constantine. Individual statues were apparently brought in under Constantius II, Valentinian, and Theodosius II.
The references given are to a publication, the Scriptores Originum Constantinopolitarum, published in two volumes in Leipzig, 1901 and 1907, ed. Preger. These are to be found at Archive.org, here and here, but … they are in Greek only. Not even a Latin translation is given.
Fortunately a shy mention in the preface indicates that a great deal of the material may be found in the Patrologia Graeca 157, col. 615 f. The volume has a useful list of contents on col. 1259, which helps with navigation.
The statues brought in under Constantius II are noted as follows (col. 720-1):
Of the statue of Zeus set up in the Hippodrome.
Many statues were sent from Iconium to Constantinople, one of which number was the Zeus which is in the Hippodrome.
Of the four gilded horses in the Hippodrome.
The four gilded horses, located on the upper side of the Hippdrome, were brought from the island of Chios when Theodosius the Younger was reigning.
Of the statues of Perseus and Andromeda.
From the city of Iconium mentioned earlier, a statue of Perseus and of princess Andromeda, who, as legend has it, and is also given by some as history, … (there follows a version of the legend of Perseus) … There (at Iconium) first Perseus and then Andromeda died, and their statues were erected above the gate of the city. In that place there were many sacrifices peformed by Decius, Diocletian and Maximian, and in the same place many saints received martyrdom. And so the statues of Perseus and Andromeda were brought, it is said, when Constantius was reigning, after the church of Antioch had been purified.
This includes the references to Constantius and Theodosius II. The four horses from Chios are those now in Venice on the roof of St. Mark’s.
However I have been unable to locate in the PG the reference give by Mango to a statue named Perichytes and one of a donkey, brought in under Valentinian:
A statue called Perichytes as well as one of a donkey, both in the Hippodrome: ibid., I, p. 64, # 64; II, p. 192f., # 82. The Perichytes was nude except for a loincloth, and wore a helmet; it was stolen by western merchants some time between 935 and 959: Vita S. Lucae Stylitae, ed. A. Vogt, Analecta Bollandiana, XXXVIII (1909), p. 39f.; ed. F. Vanderstuyf, Patrol. Orient., XI (1915), p. 1o7 ff. For other instances of the theft of statues, see Script. orig. CP. I, p. 50 (under Theodosius II); II. p. 253, # 112 (under Caesar Bardas)..
I don’t know how reliable this source is, tho, as a guide to the monuments of Constantinople. I have seen it attributed to a George Codinus, who may have lived in the 15th century, but drew on some 9th century inventory; but even this is doubtful. The text itself makes many references to earlier writers, used as sources. It might be interesting to find out!
UPDATE: I find a Bonn edition of Codinus, which contains a similar (but not identical) text here.
UPDATE: A correspondent has kindly alerted me that Averil Cameron has translated and commented on the text given in the PG, the Parastaseis syntomoi Chronikai (lit. Brief historical notes, = Scriptores originum Constantinopolitarum I, ed. Preger, p. 19-73). Cameron adds that Preger demonstrated that the text could not be by Codinus, but was 10th century. She adds that Bekker’s edition in the Bonn corpus (CSHB, 1843) and the reprint in the PG 157 are both unreliable and should be avoided. Unfortunately Preger printed no translation of any sort, and Cameron’s translation is inaccessible, so, for the general public like myself, it is PG or nothing. So be a little wary of what I have given above.
There is a cluster of related texts, all giving descriptions of Constantinople and collectively known as the Patria. My correspondent also tells me that the main work in French on the Patria is Dagron, Constantinople imaginaire, Paris, 1984; and that, in the Parastaseis, the Perichytes (= one that pours round) may be found in the note at the foot of page 61 of Preger’s edition. My thanks to him for these useful details.