Today no one follows any boundaries defined by Ecumenical councils. Ecumenical councils are important for the faith professed in these countils, but regarding boundaries within Roman empire, these boundaries are outdated and no one follows it strictly. Thus, Rome has churches in all parts of the world. Same is true about all other Byzantine Sees. Byzantine Sees received support from the Emperor, thus they gained much attention. But we cannot ignore ancient Churches outside the Byzantine empire.
There existed Apostolic Churches outside Byzantine empire. Patriarchates originated within Byzantine empire. But outside Byzantine empire ancient Churches retained their identity and Apostolic heritage - such as Armenian, Persian (Assyrian) and Indian. The heads of these Churches outside Byzantine empire were called 'Catholicos' (from 'Katholikos' meaning Universal). A Catholicos is the first among all Bishops and the administrative head of the Church. No other Patriarch can enter the jurisdiction of a Catholicos without permission.
Catholicate of the East, Catholicate of Armenia & Georgia are related to the Edessan Syrian Church. Catholicate of the East is very ancient. According to Mar Gregorios Bar Ebraya Apostle Thomas is the first Archbishop of the East - first in the lineage of Catholicos of the East. This is true both according to the tradition of Orthodox (Indian Orthodox) Church and Assyrian Church of the East. Indian Church strongly believes in the Apostolic origin. This is an important part of the heritage of the Church.
"It has long been an admitted fact that the lands of Mesopotamia and Adiabene, and in fact the whole of what we may call by anticipation the Sassanid Persian Empire, received the gospel from teachers whose head-quarters were at Edessa. The little kingdom of Osrhoene had but a precarious independence during the brief period of its existence; still that independence was sufficient to give, for as long as it lasted, a distinctive character to the Christianity that existed in its capital, and made it an appropriate "nursing mother" to the two national Churches founded by teachers who came from thence, those, namely, of Armenia and Persia. When the Edessene Church vas merged in that ecclesiastical circle that developed into the Patriarchate of Antioch, one at least of these "daughters" was strong enough to stand alone; and the circumstances of its infancy probably contributed to give it that instinct of independence that was always so marked a feature of its life.
The "Church of the Easterns" was the daughter, not of Antioch, but of Edessa, and was never included in the Patriarchate of the former city."
This explains the link between Syrian Church founded by Apostle Thomas in Edessa (mother of all Churches of Syriac tradition), The Eastern Syrian Church, Armenian Church and the Indian Church. The Antiochian Church became Syrian when a section of Edessan Church merged with Antioch. Indian, Persian and Armenian Churches maintained freedom.
Hope this clarifies the commonly held western understanding about Sees and the position of ancient Churches of the East.