During the Triduum, the question arises about when to dismiss the Elect and the catechumens. (If you are engaged in a year-round process, you will likely have both.) If parish leaders understand the liturgical role that the unbaptized have in worship, it will be clear that the Elect and the catechumens are dismissed before the priestly action of the liturgy begins.
In the Mass of the Lord’s Supper, the first action restricted to the baptized priesthood is the prayer of the faithful. So the Elect and the catechumens would be dismissed before the general intercessions. They would ordinarily be present for and participate in the washing of the feet if your parish chooses to celebrate that option.
Likewise during the Celebration of the Lord’s Passion, the first act exclusively performed by the baptized priesthood is the offering of the General Intercessions. So, just as on Sunday and at the Mass of the Lord’s Supper, the Elect and the catechumens would be kindly dismissed before this prayer.
Who leads them?
In many (all?) communities, it would be a hardship to ask a catechist to leave the worshiping assembly on these days to lead the Elect and catechumens in the dismissal reflection. And, in fact, there is no need to. The Elect will have been participating in dismissal reflections for a year or more by this point. One or more of them should have no difficulty leading the reflection sessions.
If you are practicing a year-round initiation process, you will have catechumens who will not be baptized at this year’s Easter Vigil. Should they participate in the Easter Vigil? Two pastoral difficulties present themselves in this case. The first is that the powerful symbols and ritual actions of the Vigil may have less impact the second (or third) time around, in the year the catechumens will be celebrating their own initiation. Wouldn’t it be more pastorally effective for them to experience the great Paschal fire, the Exsultet, the extensive readings and prayers of this night, and so on, for the first time on the night of their own baptism?
The second pastoral issue is who would lead them in their dismissal reflection? Unlike the Elect, the catechumens are unlikely to have a great deal of experience with this process, and it would be a great hardship to ask a catechist to be absent for the first time in which the neophytes participate in the Eucharist.
The best solution, perhaps, is to have the catechumens present at the Easter Sunday liturgy—along with the neophytes in their white robes—and to ask a catechist to lead the dismissal at that liturgy.