People should not bring up pyxes during the Communion Procession in Mass.
From the 2002 General Introduction to the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
“161. If Communion is given only under the species of bread, the priest raises the host slightly and shows it to each, saying, *Corpus Christi *(The Body of Christ). The communicant replies, Amen, and receives the Sacrament either on the tongue or, where this is allowed and if the communicant so chooses, in the hand. As soon as the communicant receives the host, he or she consumes it entirely.”
If someone is taking Communion to the sick they should go directly to them. They should not wait for a blessing at the end of Mass. But if someone is at Mass, they should not leave before the blessing. So Communion to be taken to the Sick should not be given during Mass. From the 2004 Instruction Redemptionis Sacramentum:
“[133.] A Priest or Deacon, or an extraordinary minister who takes the Most Holy Eucharist when an ordained minister is absent or impeded in order to administer it as Communion for a sick person, should go insofar as possible directly from the place where the Sacrament is reserved to the sick person’s home, leaving aside any profane business so that any danger of profanation may be avoided and the greatest reverence for the Body of Christ may be ensured. …”.