One thing to remember is that all of them were not Orthodox to begin with, yes a few but still a few had other origins. For example the Syro Malabar Church was originally considered an ecclesiastical province of the Church of the East. This province was known as the Great See of Angamli, by tradition the bishop of Angamali held the title of “Gate of India”.
The Syro Malabar Church only became in union with Rome around 1599. As you have asked, this union was triggered by the arrival of the Portuguese Catholics in India around the year 1498. Through the event known as the Synod of Diamper, the Portuguese Latin Catholic Bishop Alexio De Menezes, took claim of all the churches considered to be under the Church of the East. The Portuguese Catholics found the practices of the Native Christians ,both liturgical and traditional, to be heretical and schismatic. For this reason through the implication of the Synod of Diamper, they began the process known as Latinization (removing native ecclesiastical customs with Latin customs).
They began by burning all East Syrian manuscripts and any other texts of the Church of the East owned by the native Christians. Churches were white washed and repainted, images and icons of Isho Mishiha that gave Christ brown skin and black hair (A Middle Eastern Look) were re-imaged with Christ having white skin and blonde or brown hair. Sanctuaries were redesigned to have Portuguese Latin Catholic structures, they added things like the crucifix and icons of western saints. The last and perhaps most harmful aspect of the Latinization was that the Portuguese all but removed the East Syriac Holy Qurbana from the Native Christians, replacing it with Latin Mass. They did however allow them to retain small aspects of the Qurbana as well as continue to recite the Qurbana in Syriac.
Even through all of this, for centuries the Syro Malabar Church survived in faith and tradition. To distinguish them from the converted Catholics, European Colonizers noted the Syro Malabar Catholics as “Syrian Catholics”. In the late 1800’s through great strides by Syro Malabar priests and numerous petitions to Rome and Latin bishops, the Syro Malabar Hiearchy was restored. For the first time in centuries, the Syro Malabar Church was granted diocese’ of its own and hierarchs of its own. This gave the Syro Malabar Church the chance to grow and restore its ancient customs, which through the blessings of Christ is what happened. From the late 1800’s to today, the Syro Malabar Church began the great process of revival to its East Syriac Nature. Today the Syro Malabar Church is considered apart of the Eastern Catholic Portion of the entire Catholic Church and is the second largest of the Eastern Churches. It has 27 eparchies and a membership 4.2 million adherents in India and around the globe. I thought I’d share a little history

I hope this helped.