In many parishes, at the Passing of the Peace it is common for parishioners and even the priest himself to move about and greet individuals other than one's immediate neighbors; walking down the pew, getting out of the pew, ushers moving down the aisles greeting people, priests doing similarly, etc.
While this is undoubtedly motivated by a (mistaken) notion of fraternity and pleasantness, is it permitted according to the current rubrics?
The answer to this is a resounding no. Sometimes in liturgical matters, there is room for debate and interpretation (for example, here), but this one is cut and dry. The General Instruction of the Roman Missal and authoritative pronouncements of John Paul II and Benedict XVI all specifically prohibit this sort of activity.
First, the GIRM:
"As for the actual sign of peace to be given, the manner is to be established by the Conferences of Bishops in accordance with the culture and customs of the peoples. However, it is appropriate that each person, in a sober manner, offer the sign of peace only to those who are nearest." (GIRM, 82)
Note that while the GIRM does not dictate how the passing of the peace must be given, it does dictate to whom it ought to be given, "only to those nearest", and only "in a sober manner." This is evidently more important than the sort of sign exchanged. "Only those who are nearest" means one's immediate neighbors. The idea is to preclude a lot of moving about that would disrupt the solemnity of this particular moment in the Eucharistic liturgy.
John Paul II restates and reaffirms this discipline in his 2004 instruction Redemptionis Sacramentum. The statement of the GIRM is reiterated, with an additional reminder that this applies to priests as well:
"It is appropriate “that each one give the sign of peace only to those who are nearest and in a sober manner”. “The Priest may give the sign of peace to the ministers but always remains within the sanctuary, so as not to disturb the celebration." (Redemptionis Sacramentum, 72)
By instruction the priests to remain in place "so as not to disturb the celebration", the inverse is also proven: to not remain in place is to disturb the celebration. Therefore priests, as with the laity, ought to remain in place when offering the sign of peace.
Finally, in the 2005 Apostolic Exhortation Sacramentum Caritatis, Benedict XVI explains the mind of the Church on this matter by offering a reflection on what the GIRM and John Paul II meant when they said "those who are nearest":
"During the Synod of Bishops there was discussion about the appropriateness of greater restraint in this gesture, which can be exaggerated and cause a certain distraction in the assembly just before the reception of Communion. It should be kept in mind that nothing is lost when the sign of peace is marked by a sobriety which preserves the proper spirit of the celebration, as, for example, when it is restricted to one's immediate neighbors."(Sacramentum Caritatis, 49)
"Those who are nearest" ought to be understood as "one's immediate neighbors." It is this interpretation and this interpretation alone which "preserves the proper spirit of the celebration", and this is indeed appropriate because it exemplifies "greater restraint."
The Church clearly teaches that the sign of peace is to be extended only to those who are immediately beside us so as not to create a lot of movement that detracts from the dignity of the celebration.
Why this focus on not moving around? How does movement detract from the dignity of the celebration?
The Church wishes to restrict movement at the Passing of the Peace in order to prevent the faithful from seeing the Peace as a kind of liturgical "time out", a period of movement, talking and socializing sandwiched between two solemn parts of the Mass. The entire Eucharistic liturgy is supposed to be characterized by solemnity, and this solemnity must carry over into and through the Sign of Peace. For that reason, moving around to greet others beyond one's immediate neighbors is not acceptable, not even for a priest.
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