I can't speak to any universal norm, but there are certainly instances where the enslaved children of slaveowners would receive preferential treatment.

One notable instance is Thomas Jefferson. Of the roughly 600 slaves that he owned, he only officially freed seven - two of these were his sons Madison and Eston. They were probably 7/8^ths white although I would imagine it was more relevant that they were his children. Jefferson also allowed two more enslaved children, Beverly and Harriet, and their mother Sally's sister Critta, to escape without pursuit - he gave Harriet 50 dollars before she left and so this was certainly his way of unofficially freeing them. When they were still his slaves, he gave them light work and the boys were taught to play the violin and trained in woodworking.

Jefferson's slaves might have received better than average treatment for the time but it wasn't particularly good and he didn't do much to stop his managers from exacting brutal punishments. So it's not like his slaves were living a wonderful life of ease - which gives a bit more evidence to the idea that he was giving preferential treatment to his children and some of their mother's family.

https://www.monticello.org/slavery/slavery-faqs/property/

https://www.pbs.org/wgbh/pages/frontline/shows/jefferson/cron/1873march.html