Based on your descriptions it sounds like some bullying- so yes, I would intervene.
My DS6 had some difficulties making friends in kindergarten, but I didn't see much in the way of bullying. Some occasional kids being a little mean or snarky but not to the level of bullying. For this reason, we let him work it out a bit on his own, and tried to teach better social skills.
Asking your child to move at lunch, calling names, excluding from parties, and pushing is not okay. I tell my son that he doesn't have to be friends with everyone, but he needs to be friendly and kind. At his birthday party, he was allowed to invite only the boys or the full class. He could not exclude individuals, or guess what? No party.
A couple of suggestions:
1. talk with the teacher and ask him/her to talk with the recess monitor- you may get more insights into what happens during the day.
2. use this information to assess your daughter's skills to handle the bullying and make friends.
3. I don't know if your daughter does any extracurricular activities, but it might help her make friends in a different setting. My son takes Taekwondo at facility with a strong anti-bullying message. The benefits of martial arts have been a blessing for DS.
I don't believe that kids have to "fit in" to make friends, but some (including my own DS) could use some work on social skills and cooperation. Some assertiveness training (like my son receives in martial arts) can nip the bullying in the bud. I'm not at all talking about aggression, just some training in effective body language and words that will stop bullies.
