I know this isn’t supposed to matter if you’re a priest, but is it possible for them before or after their ordination to choose their diocese?
Religious order priests go where their superiors send them. As for secular priests, before entering seminary a man might be able to petition the bishop of a diocese not his own to consider him for the priesthood, but a bishop is likely going to be wary of a man who is “fleeing his diocese.” That doesn’t mean that a bishop might not accept a man from another diocese into his own, only that such a man might be scrutinized more carefully. Perhaps a better solution would be for such a man who is discerning the secular priesthood to move to a diocese he thinks he would like and live as a lay Catholic there for a few years before approaching the bishop about a vocation to the priesthood. After ordination, of course, a secular priest is vowed to obedience to his own bishop and generally should not be casting his eyes afield for other dioceses.