Successful Eastern Catholic churches

Successful Byzantine Catholic churches, centuries-old communities (not perfect, but successful): cases where whole patriarchates or metropolias came in. (But we're not supposed to try to split the Orthodox communion anymore, and that is good. Bring them all in together and leave the rite alone.) The metropolia of Kiev and the Byzantine patriarchate of Antioch. The Ukrainian Catholic Church at the union was the metropolia of Kiev: all of the Ukraine and Byelorussia as well (but plenty of people said no to union); Russian expansion and persecution reduced it to old Polish Galicia in the far west, still their base. The USSR grabbed it during World War II. The patriarch of Antioch who became Catholic is the basis of the Melkite Church.

The Russian and Greek Byzantine Catholic churches were originally attempts to convert the Orthodox individually that failed. (We don't do that anymore either, and that is good. We accept these conversions of course but don't solicit them.)

Looking beyond the Byzantine Rite, most Nestorians in what's now Iraq became Catholic; the Chaldean Catholic Church is bigger than the Nestorians.

And yes, it's THE Ukraine for the same reason I don't say "Paree." And the Patriarch of Constantinople is wronging the Patriarch of Moscow over there.