The (Roman) Catholic Church agrees that Peter founded different churches, like at Antioch or Rome, but it claims that it is only at the last place Peter lived or died is to where his supposed authority from Christ as the head of the apostles remains.
What is the proof of this claim? Why would it matter where Peter lived last or died at as to where his supposed authority remained?
The question is about the logical basis for the argument that the last place St Peter resided at should retain his authority as Pope.
When Peter left Antioch he did not confer his papal authority on anyone in that city, or on anyone anywhere else. He retained it personally wherever he went. It was only when he came to die that it became necessary to choose a successor. We do not know whether St Peter nominated his successor, or arranged a process for his election. But the need for a successor arose only on his death and the person who succeeded him succeeded him as Bishop of Rome and Pope, because these were the roles that his death left vacant.
It would not have been possible to summon a conclave from all over the world. Even today, notionally at least, the Bishop of Rome is elected only by the clergy of Rome, and as Bishop of Rome is ex officio Pope. The way this is done is that every cardinal, on appointment, becomes a deacon or priest or suburban bishop of a parish in the diocese of Rome itself or one of the suburban dioceses of Rome. In this way it is still the case that the Pope is chosen by the clergy of Rome.
Antioch was the first diocese St. Peter founded. Later he went to Rome, where he ruled the entire Church until the end of his life.
See St. Robert Cardinal Bellarmine in De Romano Pontifice bk. 2 ch. 4 (my emphases):
Isidore, in his life of Peter, says: “He, after founding the Church at Antioch, continued to Rome against Simon Magus under the rule of Claudius Caesar and there, preaching the Gospel, held the pontificate of the same city for twenty-five years.” Bede553 has the same, as does Freculph,554 Ado of Vienna,555 and all more recent authors.
[…] the testimonies of the ancient Roman Bishops who were martyrs or confessors[:] Pope Clement teaches that with death threatening he [St. Peter] handed on the Roman Episcopate to him.556 Anacletus in Epistle 3 teaches that on account of the see of Peter, the Roman Church is the head of all others. Marcellus I, in a letter to the Antiochenes, says: “The See of Peter was begun with you, and was transferred to Rome at the Lord’s command, etc.” Pope Damasus says that Peter was the Bishop of Rome for twenty-five years, that is, even to his death.557 Innocent I, teaches the same thing, in a letter to the Council of Miletus.558 Moreover, so do Pope Leo, Gelasius, John III, Pope St Gregory, Agatho, Adrian and Nicholas I, and all others who wrote anything, affirm that their See is the Seat of Peter.559
553. De sex aetatibus.
554. Chronicorum, tomus 2, bk 2, ch. 13.
555. Chronicum, anno Christi XLV.
556. Constit. Apostolic., bk 7, ch. 46.
557. In pontificali in Petro.
558. This is 93 among the epistles of Augustine.
559. Leo, serm. 1 de natali Apostolorum; Gelasius Epistola ad Episcopos Germaniae et Galliae; Gregory bk 2, epist. 33; Agatho in epistola ad Constantinum imperatorem; Adrian epistola ad Tharasium; Nicholas I, Epistola ad Michaelem.
See also Saint Peter and the First Years of Christianity by Constant Fouard.
source: this answer to the question "Is there any evidence to support the claim that the Apostle St. Peter founded the Church in Antioch and, if so, what are the implications?"
Not sure this should be an "answer" or part of the OP, but I'll put it here for now.
To draw out a comment from the Apostolic Constitutions, Book Vii, P 46, that is mentioned in an answer to OP, it says this (bold mine):
Of Antioch, Euodius, ordained by me Peter; and Ignatius by Paul. Of Alexandria, Annianus was the first, ordained by Mark the evangelist; the second Avilius by Luke, who was also an evangelist. Of the church of Rome, Linus the son of Claudia was the first, ordained by Paul; 2 Timothy 4:21 and Clemens, after Linus' death, the second, ordained by me Peter. Of Ephesus, Timotheus, ordained by Paul; and John, by me John. Of Smyrna, Aristo the first; after whom Stratæas the son of Lois; 2 Timothy 1:5 and the third Aristo. Of Pergamus, Gaius. Of Philadelphia, Demetrius, by me. Of Cenchrea, Lucius, by Paul. Of Crete, Titus. Of Athens, Dionysius. Of Tripoli in Phœnicia, Marathones. Of Laodicea in Phrygia, Archippus. Of Colossæ, Philemon. Of Borea in Macedonia, Onesimus, once the servant of Philemon. Of the churches of Galatia, Crescens. Of the parishes of Asia, Aquila and Nicetas. Of the church of Æginæ, Crispus. These are the bishops who are entrusted by us with the parishes in the Lord; whose doctrine keep always in mind, and observe our words. -source-
There's no sense or reason given as to why Rome would later claim it was the Seat of Peter, as opposed to Antioch, whose first bishop was ordained by Peter. For Rome, it was the second bishop. This implies that Antioch would be the head (if one agrees with this assumption in the first place).
As well, the last sentence from the quote basically confirms that all parishes are in the Lord. There's no sense of one above the others, be it Antioch or Rome.
So, OP still looking for an answer.
