When you say the mother is coming June 1st to pick up the kids, do you mean she just wants them to come to her house for part of the summer, or do you mean she wants them to live there permanently?
If it is just a summertime thing, then the father shouldn't fight against that, but if it is permanent, then he should definitely stop that from happening.
Either way, what your father needs to do is to file for custody. He will definitely win it, but if he doesn't file, it could be a lot of trouble for everybody. I know it seems right to just "go with the flow", but when things go bad, you need a court order.
It sounds a bit like my situation. My ex just left with a boyfriend and now she has moved 1000 miles away. She tried to keep it a secret too, but my daughter found out and she told me about it in front of her Mom. I didn't bother getting a court order for the first 6 years she was gone, but I regretted not doing it earlier, because when she came back she made it tough. I did get custody, but it was a lot harder than it could have been.
If your father doesn't get a court order for custody, he will regret it too. He needs to file a petition for custody of his children (other than you, of course).