St. John Chrysostom gives us a brief commentary on the greeting (Et cum spiritu tuo) And with your spirit.
"If the Holy Spirit were not in our Bishop [referring to Bishop Flavian of Antioch] when he gave the peace to all shortly before ascending to his holy sanctuary, you would not have replied to him all together, And with your spirit. This is why you reply with this expression….reminding yourselves by this reply that he who is here does nothing of his own power, nor are the offered gifts the work of human nature, but is it the grace of the Spirit present and hovering over all things which prepared that mystic sacrifice." (St. John Chrysostom, Homily on Holy Pentecost)
It seems that spiritu is a reference to the gift of the spirit the bishop/priest received at ordination. In effect we are greeting the bishop/priest acting in the person of Christ through the powers of his priestly ordination.
On the Feast of the Sacred Heart of of Jesus
SGM
