The distribution of ashes is only part of the Ash Wednesday liturgy. Official documents saying this:
“ASH WEDNESDAY
The ashes used today come from the branches blessed the preceeding year for Passion Sunday.”
(Roman Missal, Catholic Book Publishing Co., New York, 1985, page 76.)
“ASH WEDNESDAY
253 On the Wednesday before the First Sunday of Lent the faithful, by receiving ashes, enter upon the season appointed for spiritual purification. …” (Ceremonial of Bishops, Liturgical Press, 1989, ISBN 0-8146-1818-9, page 90).
“1656 The season of Lent begins with the ancient practice of marking the baptized with ashes as a public and communal sign of penance. …”
(Book of Blessings, Liturgical Press, Minnesota, 1989, ISBN 0-8146-1875-8, page 691. This section in only in the edition approved for the USA, it is not in the Latin original).
“125. In the Roman Rite, the beginning of the forty days of penance is marked with the austere symbol of ashes which are used in the Liturgy of Ash Wednesday.” (Directory on Popular Piety and the Liturgy, Congregation for Divine Worship and the Discipline of the Sacraments, 2001)
For children I can understanding having ashes to explain what they are and even giving a demonstration of what will happen on Ash Wednesday. But it would be wrong to have a ceremony on the Monday, anything like the one that should be on Ash Wednesday.
Who can distribute the ashes on Ash Wednesday?
In the USA there is the following, from the Book of Blessings:
“1659 This rite may be celebrated by a priest or deacon who may be assisted by lay ministers in the distribution of the ashes. The blessing of the ashes, however, is reserved to a priest or deacon.” (The rite being referred to is the distribution of ashes outside of Mass).
This varies from the instruction in the 1975 Roman Missal:
“The blessing and giving of ashes may be done outside Mass. In this case the entire liturgy of the word should be celebrated: entrance song, opening prayer, readings and chants, homily, blessing and giving of ashes, general intercessions.” (Roman Missal, Catholic Book Publishing Co., New York, 1985, page 78).
The 2002 Roman Missal seems to have the same thing. It could be argued that it overrules the Book of Blessings for the USA, since it is a more recent instruction. It has in Latin:
“Benedictio et impositio cinerum fieri potest etiam extra Missam. Quo in casu praemittitur Liturgia verbi, adhibendo cantum ad introitum, collectam, lectiones cum suis cantibus, ut in Missa. Sequitur deinde homilia et benedictio atque impositio cinerum. Ritus concluditur oratione universali, benedictione ac dimissione fidelium.”
Missale Romanum, Libreria Editrice Vaticana, 2002, ISBN: 8820972719, page 200. The word “benedictio” means “blessing” – the ashes are to be blessed in the ceremony outside of Mass.
I am not aware of any permission for lay people to distribute ashes outside of the USA. In the USA there is no permission for lay people to distribute ashes in Mass.
The Roman Missal has “The priest then places ashes on those who come forward, saying to each: …” (page 77)
The Ceremonial of Bishops has:
“257 After the blessing, the appointed minister, a concelebrant or a deacon, places ashes on the bishop, as the bishop bows before him, and says, Turn away from sin and be faithful to the Gospel or Remember, man, you are dust and to dust you will return.
258 Then the bishop again puts on the mitre and, seated at the chair or standing, places ashes on the concelebrants, the ministers, and the faithful. The bishop may be assisted if necessary by some of the concelebrants or deacons.”