Specific to the Ordinary Form only, my understanding is that the altar is supposed to be returned to the state it was prior to the preparation of the gifts before everyone processes out.
A server is supposed to remove the missal and stand from the altar during communion. The vessels are supposed to be removed from the altar by the servers after the priest returns to his chair. The order of events is supposed to be:
*]Priest distributes communion
*]Priest (or deacon) places unused hosts in the tabernacle. Servers should not be moving about but should stop while the priest reposes Jesus and all make an act of reverence, typically a bow of the head as the priest genuflects at the tabernacle.
*]Priest (or deacon) purifies the vessels if done at the altar. Purification may be done after mass at the credence table but not in the sacristy.
*]Priest returns to his chair.
*]Servers clear the altar
*]The concluding rites and procession then takes place.
As a practical matter, the priest must sometimes assists the children with clearing the altar prior to returning to his chair, especially if the servers are very young or if there is no deacon, acolyte or emcee. The bottom line is that the missal, stand and vessels should not be on the altar during the concluding rites or procession out. The altar should be cleared by the time the closing hymn is sung.
My information is from “Ceremonies of the modern Roman Rite” by Peter J. Elliot. I am open to correction and have no idea what happens at the Extraordinary Form.
-Tim-