Is it possible to change an adult account to a child one? My son's Apple ID has his birthday entered, cough accidentally cough, incorrectly, so it appears that he is 28, when he's actually &lt;13 (it was created when &lt;13 accounts weren't possible).

With iCloud Family Sharing, this is problematic.  I'd really like to use this, but an adult gets to purchase items without asking permission, so he needs to be a child first.

Attempting to change his birthday in the security settings in desktop Safari gives a "this cannot be changed at this time" error, and attempting to do so in the Settings app in iOS gives a "permission from a parent or guardian is required to create an Apple ID for a child of a young age" error.

Is there any way that I can correct the year of his birth?  (It would be nice to not have to create an entirely new Apple ID).
I had the same issue, I added my kids to Family Sharing but they showed up as adults (and so could make purchases without my permission). Fortunately you can easily change it. 

If you go to https://appleid.apple.com and login using their account details, in the Password and Security section, you can change the year of their birth. 1/1/2001 works fine, I think the key is that they need to be over 13 to change from Adult to Child account. I left my kids' birthdays with the correct day and month but used 2001. 

Going back to the iCloud section on my phone, it shows my kids as "Age 13".