Even though the rite is the same the cultural differences are quickly felt. In general, one priest belongs to one rite. But he can be allowed to serve the liturgy in other rites, depending upon pastoral needs and the decision of the bishops involved. In Russia, most Catholic priests are of the Roman version of the Latin rite. A few are of the Byzantine rite.
Churches are usually of the rite of their founders.
Consequently, the Roman Catholic church is Rome and all her daughter churches. They all use one of the Latin rites. The Ukrainian Catholic Church is all the Byzantine churches that were reunited with the Catholic church in the Union of Brest. They are all of Greek rite, but have a variety of sub-rites depending upon their histories. For example, if the number of Armenian Catholics in Vladivostok and the availability of an Armenian rite priest allows it, the Armenian bishop responsible for Russia can create an Armenian parish in Vladivostok, in which case the Armenian Catholics will cease being a part of our parish, and will belong to their own.