We attended a music concert for kids on the weekend and DS6 kept scowling at and making comments to a little girl sitting behind him who was being pretty disruptive.
My DD5 has a few issues too.
A few weeks ago, a boy at an event was being disruptive. When he and his mother who was clearly upset with her son left during the intermission, my DD had a very snarky expression on her face and I firmly told her that I was disappointed with her and she needed to think about why she did what she did. In retrospect, I overreacted a little bit but I was not happy that she would put another child down behind his back.
My solution has been to get her to spend more time with a friend is a wonderful model for social skills (not competitive, not bossy, asks interesting questions, thoughtful, etc. She really is an amazing 6 year old girl). After each playdate, I talk about how happy I am about their friendship and discuss details. I'm not sure how much of it is sinking in but I'm looking forward to the girls spending more time over summer.
