As stated above, the two bishops must approve. The procedure involves writing to the bishop of the church one wishes to transfer into. Provided that he approves (one should acculturate into the new church first) the letter will be forwarded to one’s bishop in the original church. Not long ago all such requests for transfer were handled by Rome, the current practice is that if the two local bishops agree the approval of Rome can be presumed.
This process is necessary to make sure that there are no impediments to a person’s good standing (all I can think of is a possible excommmunication someone is trying to hide, but some RC are not actually registered in a parish and if no record can be found the Latin bishop will not approve).
The Byzantine Catholics (these are the group I am familiar with) do not impose catechesis on registered adult Romans because it is not considered a formal conversion, so it will likely be some time before a latin rite Catholic will be used to living according to the Byzantine Calendar, fasting and worshiping as a Byzantine and understanding the spirituality as expressed there. Each bishop sets his own requirements of the time it takes to become acclimated. (I have noted one to three years). Those actually seeking to transfer should ask for instruction or sign up when it is offered.
By Byzantine Catholic I refer to the Byzantine-Ruthenian, Ukrainian, Slovak (in Canada) and Melkite churches in North America.
The letter requesting transfer must say nothing derogatory about the sister church the believer is originating from (surprisingly, some people have a negative opinion of their church’s practice or liturgy), it will surely be rejected. Likewise the local Eastern priest will probably get a call from the bishop to discuss the individuals’ level of commitment, such as to verify that the person has been living, worshipping. contributing/supporting $$ (perhaps volunteering) and confessing as a Byzantine.
No sincere individual will be denied, but it hardly makes a difference. Many Roman Catholics worship as Byzantines most of their own lives without bothering to make a formal transfer, and those individuals who come from Byzantine parents (particularly the father) will remain Byzantines (unless they request a change) even if they always worship with the Roman church, this is canon law. Thus, Saint Andrew who has posted above is actually a Byzantine-Ukrainian Catholic because of his father even if he does not identify with it, he may chose to transfer into the latin rite (although it will not change his life any) and would be required to if he was to pursue Holy Orders in the Latin church.
I know that I have written a lot here, but I hope that I have not confused anyone.