Remember when there were Stations of the Cross in Ukrainian Catholic parishes? I do not miss them; they replaced actual Eastern devotions. If they want to pray them, do it at home or at a Latin parish. I do not mind such devotions until they replace actual Eastern praxis. We are NOT just Latin Catholics with a funny Mass.The issue is longstanding and more complicated than newcomers to Eastern Catholicism think (and there is more than one kind of Eastern Catholicism). First, you don't have to adopt Latin practices to be Catholic, the point of having the Eastern rites! I'll write about the Ukrainian Catholic Church, probably the best known of the Eastern Catholic churches as it's the largest and my part-time church home. (My Byzantine Rite practice is Russian; close enough. Even my icons are from a departed old Russian man's collection. Only a few big American cities have Russian Catholic parishes. I don't talk it into the ground, and my part-time parish is Americanized, not nationalistic.) In its act of union with Rome, in 1596 at Brest-Litovsk, no latinizations, not even the filioque, were required. The church never imposed that on them. Starting about 100 years later, they and other Eastern Catholics have been latinizing themselves.
Vatican II, which are rules, not doctrine, even though it's a valid council, seemed to be open season or target practice on traditional Latin Catholicism, as the Eastern rites were told to be more traditional but not latinized anymore. Like a kind of political correctness fetishizing the non-Western. But keeping the rites unlatinized is both objectively good and a witness to the Orthodox: their practice is entirely Catholic; we don't want to destroy their culture. (Not to be confused with Orthodox anti-Westernism though delatinization can superficially resemble it.) But what of generations-old Eastern Catholic communities such as the Ukrainians, who want and miss their Stations of the Cross? The church can and should accommodate them too, up to a point. The congregation at my part-time parish didn't ask for the delatinizations and actually kind of resents them. So much for the online convert's fantasy of Eastern Catholics as closeted Orthodox resenting the evil Latins.
Which brings me to this point: often the newcomers and the locals have different ideas about the Eastern Catholic enterprise. Like most people who come from outside, I see it as a tribute and witness to the best in Orthodoxy. I tried being Orthodox, for many years; now this stuff is a personal offering from me to God through the Catholic Church. I believe everything the magisterium teaches. There are two extremes in approaches to Eastern Catholicism. You have the outsider who joins and over-identifies with the Orthodox, being rude about the locals' latinizations, lecturing them about their own church, and dissenting from Catholic teaching, such as about the Pope (which even well-meaning Catholics often exaggerate). There aren't many of them but they're noisy online; most eventually get fed up with Eastern Catholics and become Orthodox. At the other end you have the typical Ukrainian-American or Ruthenian-American who doesn't identify with or even think about the Orthodox; he wants to keep his Stations and rosary and be left in peace. That's the community I worship with, even though the priest has obeyed orders and created something that seems and is meant to seem Orthodox (the congregational plainsong version led by a diak at the kliros in back, which is great; not the Russian choral stuff, due partly to lack of resources): no more filioque, statues, Stations, or kneeling. My part-time parish sometimes sings "Immaculate Mary," the Lourdes hymn, after Liturgy; the people want it.
Yes; do the other rite's devotions at home if you want to. Rite is both to teach and keep order in church; home devotion is a free-for-all. Prayer is prayer. You can stick to one form of one rite, more than one form of one rite, mix rites, or make up your own. You can even have your own canon of saints such as dead relatives.
The church tolerates some latinizations in the parishes, lest we destroy those communities (and some including mine are endangered; aging — both Catholic and Orthodox Eastern-rite churches don't do well in America after three generations). Suppressing living generations-old devotions to try to replace them with "actual Eastern praxis" that has died out can be foolish; destructive spiritually and to the life of a community. Artificial. My approach is there should be unlatinized "Orthodox" parishes (examples are many Melkite parishes and the few Russian Catholic ones) and latinized ones up to a point; horses for courses. The Catholic Church has many cultures. I've been kicked off online Byzantine fora for defending that and for defending Catholicism. Not latinizations, Catholicism.
I don't mind latinizations in Eastern contexts as long as they're pre-Vatican II/of long standing and don't make up more than half the practice. I don't introduce them; everything I do at my part-time parish is Orthodox. But I don't talk down about them either. I shut up and listen, and "teach by example."
At coffee hour after Mass, and our people call it Mass, we don't talk about apophatic theology, having the right phronema, fasting recipes, or seeing the uncreated light. I listen to stories of the once-thriving city neighborhood they regretfully left, one of many happy Catholic neighborhoods, this one being different with the rite and, at the time, the language. We're all trying to be better people and get ready for our court appearance when we check out (I'm in my 50s; it's on my mind).