You need to have a chat w/ her mother and let her know the kind of things she is doing and saying. She may not be aware.
Ask her to put herself in your shoes. Tell her WHY you moved int he neighborhood in the first place and that you don't want your child to exclude others.
This is a great opportunity for conflict resolution skills....by parent/child and child/child and even parent/parent. HOWEVER, you may find that when dealing w/ the parent, you'll figure out why the child has the issues she has.
I would try in as many opportunities as you can, to take CHARGE of the situation. The more she's allowed to belittle, etc., the more she'll try and get by with.
In other words, invite the kids to play at your house and SET HOUSE RULES. NO negative comments, respect and kindness are the RULES. If a situation arises, use it to teach them how to resolve, compromise, etc.
DO NOT LET THIS LITTLE GIRL take control! That's EXACTLY what she's doing/attemtping to do. In addition, might want to find out more about what's going on at her nouse....NOT to be noisy......but rather, if there are some negative things going on there, this may be her way of some type of control over a situation since she can't control what's happening at home.
It's more about a LIFE SKILLS battle than it is an emotional one!! THINK LONG TERM!