M.R.
Sooooo, we have a big family with a lot of cousins and a lot of variety in parenting styles and coping behaviors.
A lot of what you describe sounds like typical gang mentality between kids, meaning they pick on the one they can get away picking on.
I recommend it's time for a big cousin pow-wow.....you and sister sit them all down together and have a big heart to heart conversation, before the next event starts, and remind them of how to play fairly, what the rules of playing together, that we learn to include everyone, not exclude, and most importantly end with the mental image...how would it feel to them to be treated the way they were treating your son???"
Ask the kids directly, "how would you like to be treated this way? How would you feel if you had the door slammed in your face when you were running after your cousins?"
I really believe it's important to get the kids to think about their choices towards their cousin. They are not making choices that create love and unity, but ones that create conflict and hurt feelings. They need to be made aware of this dynamic before it's too late.
And it's probably best you watch the kids and reward them when they learn to be nice.
Personally, I'd not allow any door slamming in my home from any kids and especially on the face of someone else.
The cousins you describe sound out of control and the sister seems OK with their rowdiness.
So, you need to step in and stop the bullying, because, and remind all these kids that it's not OK, especially at home.
If you're a religious family, you can quote the "Do unto others, as you would have them do unto you."
I have found that the above approach works well with kids who have learned some modicum of respect for their parents. Otherwise, I'd be out of there after a concerted effort to set things on a better path.
GL!
BTW - I just read your first post about how aggressive your very own son is in relating to other kids...so first and foremost, your own son has to also follow the above guidelines. Not just the other kids. He does not get a free pass on these expectations. He is going to be in a world of isolation until he learns to stop hurting others and acting out.