Desposyni according to the Church Fathers
Eusebius of Caesarea is a primary source for information about descendants of the family of Jesus. These descendants are called "Desposyni" in one of his two quotations.
In his Historia Ecclesiae, 3.20 (original text) he quotes from Hegesippus (c.110-c.180), who wrote five books (now lost except for some quotations by Eusebius) of Commentaries on the Acts of the Church, the following account, ascribed to the reign of Domitian (81-96):
There still survived of the kindred of the Lord the grandsons of Jude, brother of Jesus, who according to the flesh was called his brother. These were informed against, as belonging to the family of David, and Evocatus brought them before Domitian Caesar: for that emperor dreaded the advent of Christ, as Herod had done.
So he asked them whether they were of the family of David; and they confessed they were. Next he asked them what property they had, or how much money they possessed. They both replied that they had only 9000 denaria between them, each of them owning half that sum; but even this they said they did not possess in cash, but as the estimated value of some land, consisting of thirty-nine plethra only, out of which they had to pay the dues, and that they supported themselves by their own labour. And then they began to hold out their hands, exhibiting, as proof of their manual labour, the roughness of their skin, and the corns raised on their hands by constant work.
Being then asked concerning Christ and His kingdom, what was its nature, and when and where it was to appear, they returned answer that it was not of this world, nor of the earth, but belonging to the sphere of heaven and angels, and would make its appearance at the end of time, when He shall come in glory, and judge living and dead, and render to every one according to the course of his life.
Thereupon Domitian passed no condemnation upon them, but treated them with contempt, as too mean for notice, and let them go free. At the same time he issued a command, and put a stop to the persecution against the Church.
When they were released they became leaders of the churches, as was natural in the case of those who were at once martyrs and of the kindred of the Lord. And, after the establishment of peace to the Church, their lives were prolonged to the reign of Trajan.
—Eusebius of Caesarea, Historia Ecclesiae, 3.20
Elsewhere in "The Ecclesiastical History" (original text), Eusebius records the following account by Sextus Julius Africanus:
For the relatives of our Lord according to the flesh, whether with the desire of boasting or simply wishing to state the fact, in either case truly, have handed down the following account...But as there had been kept in the archives up to that time the genealogies of the Hebrews as well as of those who traced their lineage back to proselytes, such as Achior the Ammonite and Ruth the Moabitess, and to those who were mingled with the Israelites and came out of Egypt with them, Herod, inasmuch as the lineage of the Israelites contributed nothing to his advantage, and since he was goaded with the consciousness of his own ignoble extraction, burned all the genealogical records, thinking that he might appear of noble origin if no one else were able, from the public registers, to trace back his lineage to the patriarchs or proselytes and to those mingled with them, who were called Georae. A few of the careful, however, having obtained private records of their own, either by remembering the names or by getting them in some other way from the registers, pride themselves on preserving the memory of their noble extraction. Among these are those already mentioned, called Desposyni, on account of their connection with the family of the Saviour. Coming from Nazara and Cochaba, villages of Judea, into other parts of the world, they drew the aforesaid genealogy from memory and from the book of daily records as faithfully as possible.
—Eusebius of Caesarea, Historia Ecclesiae, 1.7.11,13-14