What does “prōsit” mean?

After the Eucharistic celebration, the celebrant, concelebrants and servers move in procession toward the sacristy during the recessional hymn. All bow to the processional cross, held by the cross bearer who (together with the candle bearers) turns to face the celebrant and the other servers. The celebrant may use a customary formula signifying that the liturgical action has been accomplished. According to Roman customs, he may say “prōsit”, bless the servers before thanking them and remove the vestments.

“Prōsit” is a Latin verb conjugated in the third person singular of the Present Subjunctive of the verb “prōdesse”, a variant of the verb “esse” that uses an epenthetical “d” because this form of “sum” begins with a vowel (ie prōdes, prōdero, etc.) “Prōdesse” means to profit, to benefit, to be advantageous (to) (+ dative case). Its principal parts are "prōsum, prōdesse, prōfui, profuturus". 

Therefore, “prōsit” literally means:

1. "May it profit [us].” or "May it be of profit [to us].”

2. “May it benefit [us].” or "May it be of benefit [to us].”

3. “May it be advantageous [to us]."

Since it pertains to the liturgical action that has been accomplished for our benefit, the concelebrants may also respond “Prōsit” while the servers’ response is “Per omnibus et singularis.” (for all and for each.)

Sources :

Collins, John F., A Primer of Ecclesiastical Latin, Washington DC: The Catholic University of America Press, 1985.