Not all of the Eastern Churches are Byzantine.
The Armenian Catholic Church is Armenian Rite
The Chaldean, Syro-Malabar, Syro-Malankar, and Maronite Churches are Syriac Rite
The Coptic and Ethiopian Churches are Alexandrian Rite
There is at least one of the Antioch rite
Most of the REST are Byzantines…
The Most notable Byzantines:
The Ukrainian, The Ruthenian*, the Melkite**, Russian***
*The Ruthenian Church is darned near two ritual churches… AMerican and European. European Ruthenian Jurisdictions are direct papal dependencies, and not sui iuris; the American Metropolitan Church is sui iuris, but has no authority sharing other than the pope with the European Ruthenian Hierarchs… But, a Ruthenian from one side is canonically enrolled in the other as well…
** The Melkites are of the Antioch rite in origin, but are byzantified. They are fundamentally Byzantine, and from what I gather, the Rite of Antioch is the genesis of the Rite of Constantinople (the Byzantine).
*** Notable for the absolutel lack of bishops and yet being a major thorn in the Orthodox-Catholic dialogues. who knew than fewer than a dozen parishes worldwide could be SUCH a problem?