The timeline is huge, lets start from the very beginning. Under the appropriately named Papa Mar Gaggai bishop of Selucia the bishops of the Persian empire started to organize themselves into a Synod. By 410 under the Metropolitanate of Mar Issac a synod was held and the Bishop of Selucia became Grand Metropolitan and Primate of the Church in the East, Mar Marutha of Martyropolis (Miapharqin) deputed by the Byzantine Emperor Arcadius attended the Synod and led a Roman delegation to the Persian court of Yadzegard I ( Almost all major secular hsitorians hold Mar Maruta to be the delegate of the Emperor and not of the Antiochian Patriarch) . In the Synod of 420 under Mar Dadisho autocephaly was declared by the Bishops in the East.
Then with the execution of the Orthodox Catholicos Babowai, the See of Selucia was usurped by the Nestorians led by Barsauma Bishop of Nusaybin and thus seperated from the rest of the Orthodox.
(From now I am depending of a book written by the present Syrian Orthodox Patriarch)
After the execution of Babowai and the acceptance of the teachings of Theodore, Diodore and Nestorius except for a minority in Norther Iraq (centered around Tikrit, Nineveh, Erbil, Mosul and frontier areas ) the whole of the Eastern Church accepted Nestorianism.
After the death of Barsauma of Nisibis, the Armenian Catholicos Christophoros visited the East and consecrated Monk Garmai as a bishop in the Monastery of Saint Mathew and gave him authority to consecrate bishops, as the Catholicos of the East. Christophorus also consecrated Monk Ahodemeh
bishop in Baerbye(In the region of Bartellah, Iraq) . In 559 AD, Mar Yacoub Baradaeus (General Metropolitan of the Non Chalcedonians) visited the church in the East and consecrated Ahodemeh as General Bishop (Catholicos) for it and was considered the first General Bishop of the East, after the Nestorians had captured its see.In 628 AD, a reconciliation was reached between thePersian and Roman Empires. Patriarch Athanasius I (595-631) sent his secretary Rabban (Monk) Youhanna to theEast. He met with Bishop Christophorus, head of theMonastery of Saint Mathew and discussed with him the subjectof resuming relations between the See of Antioch andthe Church in the East. The bishop convened a synod, whichwas attended by Monk Yauhanna and four regional bishops.They elected three monks, Marotha, Ithalaha and Aha, and requested the patriarch to consecrate them bishops. The patriarch regretted in order to preserve the old custom of the Church of the East, which in the absence of the catholicos, and in case of an unavoidable necessity, three bishops together can consecrate a new bishop. Then the Eastern bishops, and in the presence of the patriarch’s bishops consecrated the chosen monks bishops. The patriarch installed Marotha, one of the three new bishops, as Bishop (Catholicos) of Tikrit, and gave him authority to preside over the East, on his behalf.
Mar Marutha of Tikrit became Catholicos and also took the title of Maphriana , primarily to distinguish the Orthodox Catholicos from the Nestorian one.
I accept the significant parts of the Patriarchal narrative, but it is also true that beginning with Igantius Afrem Barsoum, an effort was made to advance Patriarchal claims. The Syriacs wrote a lot of histories, Dionysius Tel Mahre, Zechariah the Rhetor, the Zuqnin Chronicles, the history of Micheal the Great, that of Catholicos Bar Ebroyo etc, to properly understand the dynamics of the relation between the Patriarch at Deir Mar Hananiyo and the Catholicos in Tikrit or at Mar Matta Monastery a study of such primary sources is crucial. Needless to say, I have not done that.
Uptil 628, perhaps the liturgy used by those in Iraq was the one of Edessa used by the Nestorians also, after 628 perhaps the rite of Antioch was used. Hoever even today the Eastern orders of Mosul and Tikrit are different from those used by the Syriacs in the Turkish mountains. The orders of Edessa also have significant differences.
The Patriarch continues below :-
Mor Marotha of Tikrit (d. 649 AD) was the first to be called Maphrian and from him the Maphrianate took its succession. It is worth mentioning that the bishoprics of the East increased in number and prestige to the extend that they outnumbered the dioceses of the See of Antioch, during the
time of Mor Gregorios Bar Hebraeus who was the Maphrian of the East (1264-1286), as he himself declared. Bar Hebraeus is considered as one of the most famous scholarly Maphrians of the East. The headquarters of the Maphrianate was first in Tikrit and remained there until 1089 AD. Subsequently, it was
transferred to Mosul, then brought back to Tikrit until 1152, when it was transferred to Saint Mathew Monastery. For some time, the Maphrianate was moved to Barttleh, near Mosul and then to Mosul itself In the past, it was the custom to have the Maphrian keep his episcopal name, even after his installation. However, from the 16th century onwards, it was decided to have the name Baselios added to his original personal name. In the year 1860, after the death of Maphrian Mor Baselios Bahnam IV of Mosul, the Maphrianate was abolished by a decision of a synod ( the Syriac synod).
At the zenith of its influence, the Catholicos Maphrianas and his bishops held the Synod of Kapurthatha in 860 whose canons are listed by Catholicos Bar Ebroyo and the canons regulate the relations between the See of Antioch and that in the East. The relations between the two Sees was not exactly perfect and disputes and temporary schism were common.
However after the Mongol invasion of Iraq , almost all the dioceses of the East are laid waste. Tikrit is cleaned of Christian precense with both Orthodox and Nestorian churches destroyed ( incidentally under terms agree between the Nestorians and Orthodox, a Church for the Nestorians was built in Tikrit while one for the Orthodox was built in Bagdad). After this the dynamics of the relationship is definately different. The Maphriana is now a functionary under the Patriarch with not many pretensions to autonomy. A number of Maphrians eventually became Patriarchs and both the Eastern and Western Dioceses became one administrative unit. Inspite of this, Maphrianas till the 17th century like Mar Baselios Yeldo referred to themselves as Catholicoi in their service books aware of the canonical background.
What relation did the Indians have the Orthodox Catholicoi of the East. Bar Ebroyo was one person who would have known, afaik he does not give an reference. Between the 13th to the 16th century, in India also no such links have been found. After Mar Gregorios Abdul Jaleel of Jerusalem, a number of links exist, and the Indian liturgical rite closely follows that of West Syrian -Eastern orders of Mosul .