The majority of Roman Catholic missionaries both priests and nuns are Eastern Catholics of the Syro- Malabar rite of Kerala (East syrian/Chaldean rite). They are absorbed into the Latin Rite, and all though they are said to be biritual religious, they basically become Latin rite priests under Latin Rite bishops.
The Orthodox ofcourse are under no such restrictions, but Catholics over the last 200 years have built up very efficient missions and associated centers. I must confess that in comparison to the Catholic effort, the Orthodox lag behind.
The Syro Malankara Catholics who went over from the Orthodox, are still in 'unionist' mode in Kerala and try to get Orthodox to come over, it mostly happens through a number of educational institutions that have been built by them, in return for education and employment oppurtunities many Orthodox are encouraged to go over.
Malankara Catholics also try and build churches close to Orthodox ones and since the liturgy is the same, they try to get Orthodox to go over.
I am not being polemical , but this is the common Orthodox perception, and relations are strained between the Churches often.
On the other hand relations with the other Eastern Catholic church in India, the Syro Malabars are considerably better, because there is no attempt for proselytization aimed at the Orthodox.