I heard that the priest is supposed to lead the procession at a wedding. Is that true?

Yes. In the Rite of Marriage, the ministers go first followed by the priest and then the bride and groom.
If there is a procession to the altar, the ministers go first, followed by the priest, and then the bride and the bridegroom. According to local custom, they may be escorted by at least their parents and the two witnesses. (Rite of Marriage #20)

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