Dear Friends,
The issue of the adoption of religious Orders from the West can truly fill several books!
Met. Andrew Sheptytsky did indeed allow for the establishment of several Western Orders in the UGCC to fulfill certain functions and roles.
The Oblates of Mary Immaculate are a case in point and the Belgian Father who established the Byzantine section of this Order (founded by St Eugene de Mazenod of Aix-en-Provence and Marseilles - I visited his chapel in June!

).
He built many Byzantine Catholic churches in Western Canada that are still there today and the Oblates are still active in the western provinces of Canada.
Then there were the Redemptorists, four of whose number have been Beatified and Bl. Basil Velychkovsky's relics and shrine are today located in Winnipeg, Canada.
They preached missions and I am grateful to them for the "Redeemer's Voice" publication and Fr. Michael Schudlo's (+memory eternal!) prayerbooks that nourished me spiritually as I grew up.
The Basilians were reorganized by the Jesuits and they have a long history where RC's would enter their ranks to spread the Union among the Orthodox etc. The Basilians today are called the "Basilians of St Josaphat."
Interestingly, during the life of St Josaphat, someone suggested establishing a Byzantine Carmelite Order - and St Josaphat vehementaly opposed the move!
He believed importing Western orders would hurt the cause of East-West reunion etc.
And he believed the only real monastic spirituality for the East was the Rules of ST Basil.
He once travelled to Kyiv to vist the Kyivan Pecherska Lavra to venerate the relics of the more than 150 monastic Saints there (plus the skulls of 61 unknown myrrh-bearing Saints).
As he entered the gate of the Lavra, he was recognized by one of the monastics, on his way out to do some hunting, set upon him, punching and kicking him, shouting, "So you 'soul-snatcher' have finally come here to do mischief, have you?"
Josaphat calmly told him that he did not come to cause trouble but to venerate the relics of the Pechersky Saints.
The angered monastic calmed down and let him pass.
Josaphat turned around and called to him, however, saying, "But where is it stated in the Rules of our Father, St Basil, that a monk may hunt?"
Clearly, he saw himself as part of the same monastic tradition of Orthodoxy.
Unfortunately, the UGCC has had more problems with its (albeit Latinized) Basilians, descended as they are from the original Orthodox monastic tradition, than with the members of the later Byzantine houses of Western religious Orders . . .
Alex