God’s people were transformed from conducting a pilgrimage as a devotional exercise to being challenged to make pilgrimage as a way of life. If we are urged to imitate Christ, then we cannot deny we are urged to be pilgrims as he was a pilgrim:
281. When Jesus accomplished in himself the mystery of the Temple (cf. John 2,22-23) and had passed from this world to the Father (cf. John 13,1), thereby going through the definitive exodus in his own person, no pilgrimage was binding any longer on his disciples: their entire lives now become a pilgrimage towards the sanctuary of heaven and the Church is seen as an “earthly pilgrimage”(Roman Missal, Eucharistic Prayer III, Intercessiones).
Pilgrimages are not just a Catholic thing:
The Church, however, because of the harmony between her teaching and the spiritual values inherent in pilgrimage, has not only regarded pilgrimage as a legitimate form of piety but has encouraged it throughout her history.
The Directory on Popular Piety and the Liturgy is online at the Vatican site.