Pretty much the same way their ancestors first submitted to the Vatican: it seized their Orthodox Churches.
It looks to me like it is necessary in some cases, but not in all.
Is it necessary upon reception into the Orthodox Church to renounce the errors of Yes.
For example, if I am not mistaken, there was a situation in the Ukraine after WWII, when the Eastern Catholics were received into the Orthodox Church, simply by going to the same Church they had been attending, but now it was an Orthodox Church.
The bishops of Rus' agreed to Unia in 1596, but there was room for dissent.
The lacky of the King of Poland, the apostate Metropolitan of Kiev, closed all the Churches in Brest, so the Faithful Orthodox bishops could not meet when the schismatic bishops met to submit to the Vatican. As Orthodoc summarized best:
Ref: The Council met in the city of Brest on October 6, 1596. In order to prevent a parallel Orthodox council in any of the numerious Orthodox Churches in the city, the now Uniate Metropolitan of Kiev sealed all Orthodox Churches on the day before the Council was to begin, except for the cathedral where the Council was to take place. The Orthodox, nevertheless, converged on Brest as well, with prince Ostrozhskii and his private army at the head. Failing to find an open church, and after waiting in vain for an invitation from the Uniates, they accepted an offer of a Protestant church school for a separate Orthodox Council. The Uniate Council passed a resolution excommunicating all the Orthodox clergy and laity participating in the Orthodox Council. The Orthodox in turn suspended all the clergy and lay participants in the Uniate Council and addressed a petition to the King, asking him to deprive "the traitors" of their dioceses and parishes. But the Polish King decided otherwise: his edict of October 15, LEGALIZED ONLY THOSE BYZANTINE RITE CHRISTIANS WHO JOINED THE UNIA; IT DECREED THE ORTHODOX CHURCH NULL AND VOID AND ALL ITS CLERGY EXCOMMUNICATED; WHILE CONTINUING MEMBERSHIP IN THE ORTHODOX CHURCH WAS DECLARED TO BE AN ACT OF TREASON AGAINST THE STATE.
The dyzunici existed, no one persecuted the small minority which maintained the schism.
Why would they persecute them? The small minority who went into schism by submitting to the Vatican were the only ones the King and the Sejm recognized. The vast majority who remained faithful to confessing the Orthodox Faith of the Catholic Church were the ones who persecuted, to make them match their official nonexistence.
The Polish-Lithuanian-Rusin nobility allowed Orthodoxy and Greek Catholicism to exist side by side. There were two metropolitanates Orthodox and Greek Catholic.
Only after 1632. The Patriarch of Jerusalem (the real one, not the Crusader usurper) consecrated new bishops in the PL Commonwealth, which had banned the entry of foreign clergy, and hunted down the consecrated. The Cossacks, however, made sure the king's warrent was never executed. When the Sejm and King broke down to the reality of the utter failure of Brest-Lvov, however, they refused to recognize these bishops and they did not get the rights the new, legal, Orthodox bishops had.
A History of Russian Christianity Volume II the Patriarchal Era Through Tsar Peter the Great 1586-1725 By Daniel H. Shubinhttp://books.google.com/books?id=5KHmOxc_bswC&pg=PA147&dq=1620+Peter+Mogila&hl=en&ei=88z_Tc-9F8T40gGa_4zZAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q=1620%20Peter%20Mogila&f=false
The Russians and their church By Nicolas Zernovhttp://books.google.com/books?id=JTqWtaoxAZMC&pg=PA87&dq=1620+Peter+Mogila&hl=en&ei=88z_Tc-9F8T40gGa_4zZAw&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEkQ6AEwBg#v=onepage&q=1620%20Peter%20Mogila&f=false
In 1839 and 1946, political commissars liquidated the Greek Slavonic Catholic Church.
It had already, given that the bribe factor was gone, began to dissolve. The political commissars came only in the wake of such saints as Alexis Kabaliuk the Confessor
St. Maxim Sandovich the Neomartyr
St. Vasily Martysz the Hieromartyr
who continued the work of St. Athanasius of Brest-Litovsk
There was no room for dissent from the policy of the Russian and Soviet states.
Nor in the late Polish-Lithuanian Commonwealth (which is how it met its demise) nor the Second Polish Republic.
The Lutheran Swedes, Finns, Estonians, Latvians and Baltic Germans, the German Mennonites etc. or even the Ultramontanists Lithuanians and Germans had nowhere near the problems the Ultramontanist Poles made for themselves. Pride comes before a fall, and arrogance prepares destruction.
Officially, the Russians and Soviets delegalised the Greek Catholic Church.
The Vatican and its polish minions showed how to do that.
The Greek Catholic metropolitanate of Kiev ceased to exist.
It existed only on paper.
No room for pluralism in the Russian/Soviet state.
The existence of the Lutherans and other Protestants, the Muslims, and even loyal Ultramontanists belies that. It is Ultramontanist states that can't take competition.