At this time, the only canonical Church that has "ewestern rite" parishes is the Antiochian Orthodox Church. Two versions of the "western rite" have been approved:
1) The "Divine Liturgy of St. Gregory the Great" which is an English translation of the Tridentine Latin Rite with some "corrections" and some prayers borrowed from the Byzantine Liturgy, and an Epiclesis of the byzantine rite added.
2) The "Liturgy of St. Tychon", which is a "corrected" version of a conservative Anglican Prayer Book with numerous prayers borrowed from the byzantine rite, and even from the modern Catholic Mass.
It is very true that the supporters of the "western rites" have rightful intentions in trying to reconcile those christians who were alienated because of liturgical abuses in their former parishes, to the Orthodox Church, but for many authors, the "restoration" of western liturgies has a problem of legitimacy and practice. According to father A. Schmemann, the liturgy currently used at the Western Vicariate, is not the pre-schism liturgy in its fulness, as some subtle modifications to the Latin Mass were done immediately after the schism, and then in Trent as a reaction against Protestantism (two moments of severe crisis in the West). But the most serious thing is what happened many centuries later, in our times.
It can be said that even if the liturgy celebrated in the WRV, although not 100% the pre-schism mass, is the fruitful tridentine liturgy, that fostered holliness, serenity, vocations and sanctity to the Latin Church. However, the liturgy of the WRV is, most likely, not the Tridentine Mass either. What they have there is in fact a translation of the 1962 Missal of John XXIII. It is important to notice that this Missal already contained changes such as the supression of collects, the last gospel and the prayers of the altar on many ocasions; the conmemoration of saints made optional on sundays and lower ranking fests; and modifications in the mass propers and the services of Holy Week. This transitional missal became obsollete only three years after its promulgation, before it was replaced by Paul VI's mass.
It would be difficult to say what liturgy is really followed in the WRV because, the rubrics are a mishmash of Byzantine and Western elements, traditional and modern. The whole Liturgy is celebrated in English, just like in the last year of the 1962 Missal. Many customs have been borrowed from the Episcopalians but were already present in the Missal, such as lay readers and priest facing the congregation, and protestant hymns. Elements from the byzantine liturgy such as the veneration of icons and some traditions have been incorporated, and "native" elements have appeared such as the very eclectic way in which communion is given.
According to Fr. Schmemann, Eastern Orthodox would probably not feel confortable and would abstain for communion, a tridentinist would not agree with those notorious changes, and it wouldn't be the way modern anglicans and catholics worship. The Orthodox Bishops of other jurisdictions were very critical. The Greek Orthodox Bishop of St Francisco issued a letter discouraging their faithful to attend those parishes and forbiding the concelebration with those priests.
If the first liturgy has some problems, the second one, named after St Tychon is seriously defficient. First of all it comes from a Protestant Church and we have no doubts that Protestantism has nothing to do with Orthodoxy. Since its creation, that liturgy was filed with the spirit of Protestantism. It had to be corrected with Russian Orthodox liturgical texts in a way, that it looks a little bit like an abreviated Liturgy of St John Chrisostom, but apparently, some prayers from pope Paul's mass were added to it in order to "connect" the texts.
Many western rite parishes, after seeing the inviability of their work, have turned inti byzantine ones. There's also the belief that the western rite would be seen as a kind of "reverse uniatism", and not as a helpful way to reconcile East and West.