In most shuls that I have attended (mainly Nusach Ashkenaz), young children (below Bar Mitzvah age, usually) finish the Shabbat davening at the end of Musaph. I.e., they sing from En K'elokeinu through Adon Olam.
How and why did this custom begin?
I have a concern with their doing this. While this is not a scientific poll, I asked about 20 young kids (ages 6 - about 11) if they actually recite the entire Pitum Haktoret, the passage after En K'elokeinu. All said, no. The words were too hard from them to pronounce, and I guess, being that there is no song to them, I can understand, somewhat, why this is harder than "Aleinu". OK, if, technically, you're a Shaliach Tzibbur, aren't you supposed to say everything? Is it correct to assign a kid who can't do it and just says the ending "for show"?
I can probably say the same thing about An'im Zemirot, though, it seems that since they have to sing it aloud, that may be an incentive for them to pronounce the words better. (When I was a younger boy (not a "Yunger man"), my chazzan would not let me sing Anim Zemirot, exactly because of this concern that the words were too hard to pronounce.)
In short, a combo question - why did he custom become this way, and is it the correct thing to use kids that can't pronounce the words or don't even bother in the first place?