Alexandria and Antioch actually split off from Constantinople about six centuries before the Rome-Constantinople split in 1054 (note that it is not clear that there actually was a full split in 1054 - see usccb.org/beliefs-and-teachings/ecumenical-and-interreligious/ecumenical/orthodox/filioque-church-dividing-issue-english.cfm).
Alexandria split off following the Council of Chalcedon in AD 451 because Alexandria rejected the doctrine of the dual natures of Christ, teaching instead that Christ had one nature (monophysitism - the heresy of Eutyches, who claimed that the Logos occupied the highest part of Christ’s soul, so Christ was missing part of his humanity), or as they claim today, a “compound nature” of God and man (miaphysitism - how this differs from Chalcedon I really don’t understand, it may be a language issue). Alexandria appointed their own Patriarch Timothy II in AD 457, while the Byzantine Empire from Constantinople tried to force their own Patriarch, Proterius.
Antioch has a more complicated history, because its region, Syria, was divided not only between Chalcedonians and monophysites, but also with Nestorians, who claimed that Jesus was two persons rather than one person (they would probably dispute that, again there are a lot of language issues in a world where one see was speaking Latin, another Greek, and another Syriac). And there were also orthodox believers in Syria, the Maronites, who always remained loyal to Rome. Meanwhile, the Byzantine Empire from Constantinople tried to force its own Patriarch on Antioch, so in the decades and centuries following the Councils of Ephesus and Chalcedon, there was actually a 4 way split in Antioch:
I haven’t been able to find as much information on Jerusalem, most likely because it was a much smaller church compared to the other 4 (Rome, Alexandria, Antioch and Constaninople) and didn’t have much influence outside its own limits, but it also appears to have split in a manner similar to Antioch, since it was roughly in the same region as Antioch
These divisions within the eastern church go back further than Chalcedon and Ephesus. When the church was still trying to overcome Arianism, Saint Jerome noted:
Since the East, shattered as it is by the long-standing feuds, subsisting between its peoples, is bit by bit tearing into shreds the seamless vest of the Lord, woven from the top throughout, since the foxes are destroying the vineyard of Christ, and since among the broken cisterns that hold no water it is hard to discover the sealed fountain and the garden inclosed …
Saint Jerome contrasted the rampant schism and heresy in the east with the unblemished orthodoxy and unity of the Church of Rome:
I think it my duty to consult the chair of Peter, and to turn to a church whose faith has been praised by Paul. I appeal for spiritual food to the church whence I have received the garb of Christ. The wide space of sea and land that lies between us cannot deter me from searching for the pearl of great price. Wheresoever the body is, there will the eagles be gathered together. Evil children have squandered their patrimony; you alone keep your heritage intact. The fruitful soil of Rome, when it receives the pure seed of the Lord, bears fruit an hundredfold; but here the seed grain is choked in the furrows and nothing grows but darnel or oats. In the West the Sun of righteousness is even now rising; in the East, Lucifer, who fell from heaven, has once more set his throne above the stars. Isaiah You are the light of the world, you are the salt of the earth, you are vessels of gold and of silver. Here are vessels of wood or of earth, which wait for the rod of iron, Revelation and eternal fire.
- Yet, though your greatness terrifies me, your kindness attracts me. From the priest I demand the safe-keeping of the victim, from the shepherd the protection due to the sheep. Away with all that is overweening; let the state of Roman majesty withdraw. My words are spoken to the successor of the fisherman, to the disciple of the cross. As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the rock on which the church is built! This is the house where alone the paschal lamb can be rightly eaten. This is the ark of Noah, and he who is not found in it shall perish when the flood prevails. But since by reason of my sins I have betaken myself to this desert which lies between Syria and the uncivilized waste, I cannot, owing to the great distance between us, always ask of your sanctity the holy thing of the Lord. Consequently I here follow the Egyptian confessors who share your faith, and anchor my frail craft under the shadow of their great argosies. I know nothing of Vitalis; I reject Meletius; I have nothing to do with Paulinus. He that gathers not with you scatters; he that is not of Christ is of Antichrist.