Why is the Last Place on Earth St. Peter Founded the Seat of his Authority?

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?