From the perspective of Roman Catholicism, what do Orthodox churches have to do to become "Eastern Catholic Churches"? Recently I answered a question and included an image that attempts to portray, among other things, the relationship between the Eastern Orthodox, Oriental Orthodox, and Eastern Rite Catholic Churches:



This image shows churches leaving other branches and joining the Catholic Church, specifically, the Eastern Catholic Churches.  I understand from Wikipedia that there are 23 such churches "in full communion with the Pope in Rome," including the Syriac Catholic Church.  Wikipedia is vague on the details, but says that in 1782 the Syriac Patriarch "declared himself Catholic and in unity with the Pope of Rome," which resulted in a division of the Syriac Church into Catholic and Oriental Orthodox wings.  

My question relates to how this "in unity" process works, specifically regarding national/ethnic churches of the Eastern Orthodox or Oriental Orthodox Churches.  An overview is fine – I don't need details if it's a complex process – but I'd like to understand what the Catholic Church expects from such church leaders who attempt to bring their Orthodox churches into the Catholic Church.  

In case it's helpful, I'll mention some ideas that come into my mind, but I don't know if these examples make sense.  Do these church leaders simply have to declare their unity with the Pope?  Do they have to explicitly reject some other authority?  Does a unification ceremony of some kind have to take place?  Do church leaders have to demonstrate somehow that their action represents the will of their followers?  Have standards changed over the years?
The Vatican II document which talks about the Eastern Churches is Orientalium Ecclesiarium. In it, the Conciliar Fathers turn their attention to these Churches and their Orthodox counterparts.

In particular, the Church deems it very important that the Eastern Churches maintain their own liturgical and spiritual tradition:


  
  All members of the Eastern Rite should know and be convinced that they can and should always preserve their legitimate liturgical rite and their established way of life, and that these may not be altered except to obtain for themselves an organic improvement. All these, then, must be observed by the members of the Eastern rites themselves. Besides, they should attain to an ever greater knowledge and a more exact use of them, and, if in their regard they have fallen short owing to contingencies of times and persons, they should take steps to return to their ancestral traditions.
  


In particular, latinisation (and byzantinisation, as well) is to be avoided where possible, and in fact much work has been done to remove accretions to the Eastern Rites that have come from the Western ones, sometimes to the dismay of the people who have grown accustomed with those.

Regarding the requirements for Orthodox Churches to join the Catholic one, they are mentioned in §25:


  
  If any separated Eastern Christian should, under the guidance of the grace of the Holy Spirit, join himself to the unity of Catholics, no more should be required of him than what a bare profession of the Catholic faith demands. Eastern clerics, seeing that a valid priesthood is preserved among them, are permitted to exercise the Orders they possess on joining the unity of the Catholic Church, in accordance with the regulations established by the competent authority.
  


To a Chalcedonian Church, this "bare profession of Catholic faith" involves basically recognising the authority of the Holy Father. The Union of Brest, by which the Ruthenian Church came into union with Rome, stipulates that the Eastern Catholics need not insert the filioque clause into the Creed, they need not accept the doctrine of Purgatory (though they may not explicitly deny it), and need not hold fast into any of the elaborations that the Western Church has performed upon Transubstantiation (it is enough to believe in the Real Presence).

To a Miaphysite Church, they need to accept furthermore a Chalcedonian Christology, even if only by accepting that what the Miaphysites call φύσις (physis, nature) the Chalcedonians call ὑπόστασις (hypostasis, being).

Regarding the Church of the East, it's not clear that they have a truly Nestorian Christology, though they honour Nestorius. The records of the Synods of the Bishops of the Sassanid Empire during the fourth and fifth Centuries show a studious silence on the question of the natures and hypostases of Christ, and the schism of the Eastern Church was more a question of the needs of the Persian Church to distance itself from the Roman Emperor than an actual question of theology.

In summary, there is very little that the Orthodox would need to do to restore unity with Rome. Foremost would be to accept the primacy of the Holy Father, and most of the rest can be left to the discretion of each Church. Again from Orientalium Ecclesiarium:


  
  History, tradition and abundant ecclesiastical institutions bear outstanding witness to the great merit owing to the Eastern Churches by the universal Church. The Sacred Council, therefore, not only accords to this ecclesiastical and spiritual heritage the high regard which is its due and rightful praise, but also unhesitatingly looks on it as the heritage of the universal Church. For this reason it solemnly declares that the Churches of the East, as much as those of the West, have a full right and are in duty bound to rule themselves, each in accordance with its own established disciplines, since all these are praiseworthy by reason of their venerable antiquity, more harmonious with the character of their faithful and more suited to the promotion of the good of souls.
  


Thus the Churches of the East can and must enjoy a large degree of autonomy.