It says in OC Siman 339 Sif 4 that judgements are not made on Shabbos. As well the Rema writes there that we therefore don't put someone into jail on Shabbos. As well we don't hit them since this is considered to be part of making "judgements" on Shabbos. The M"B there in note 12 says that as well one could come to make a mark on the person which would be a problem on Shabbos.
What about in regards to dealing with a misbehaving child on Shabbos. May one "punish" the child by hitting them (putting aside any discussion of whether it's right or wrong to hit a child -- asking only in regards to Shabbos.) Obviously not hard enough to make some sort of mark (as stated above -- which anyway a person would want to avoid hitting a child so hard.) However in a case where it's a light "poch" if this would be muter on Shabbos (in light of what it says here in Shulchan Aruch.) Are there any achronim that perhaps spoke about this?
(I suppose that one could ask not only in regards to hitting but other "punishments" as well, but I decided to use hitting as the main example since it's discussed here as well in Shulchan Aruch and is ossur because it's in the category of making a judgement which is ossur on Shabbos.)