Of course he has rights.  He has the right to his children.
If she wants to go...GO...but both children stay and he should contact a lawyer right now to make it so.
Also, where is she getting the money for this trip?  If it's from him or his account or a joint account, clean them out NOW.  Let the new guy pay for the trip.  Any past promises made about money are now null and void as they don't relate to this kind of situation.

You're conflating two issue here.  First you're worried about the "emotional affair", which is completely moot.  It's also moot about where she got the money.  Children are a completely separate issue.  Given that neither your brother nor the woman seem to have a high sense of Catholic morals, is it better of the children were to stay with him vs. with the woman.  In the court, there would likely be some joint custody situation, since the courts acknowledge "common law" relationships.

But he is required to separate from living in sin with here ANYWAY, regardless of this "emotional affair".

I think as a legal parent he can get an emergency court order to stop her from taking the child out of the country- it's a start anyway.

Children are victims of your brother and the women he's with, but none of this changes the fact that he can't continue living with this woman, and the "emotional affair" is meaningless, in that he has not right to any kind of more-than-purely-Platonic relationship with this woman.

This is the OP.  There are other anonymous people posting in this thread.  (I don't know why they want to hide their identity. My reasons should be clear.)  I did not write the post you are replying to.  I agreed with your take on the situation and upvoted you.

I think the problem here is that too many Catholics have become desensitized to the seriousness of these immoral practices. 

This is the OP.  Having a front row seat to this situation has given me a new appreciation for the wisdom of Church teaching.  While the world complains about Catholicism lacking in compassion, look at all the suffering that happens when people ignore Catholic moral teaching.

I don't have much sympathy for the adults in this situation, who are facing the consequences of their actions, but the children are innocent.  I feel sorry for them. 

This is the OP.  Having a front row seat to this situation has given me a new appreciation for the wisdom of Church teaching.  While the world complains about Catholicism lacking in compassion, look at all the suffering that happens when people ignore Catholic moral teaching.

I don't have much sympathy for the adults in this situation, who are facing the consequences of their actions, but the children are innocent.  I feel sorry for them.

Yes, I agree.

She can't travel with his child out of the country without his permission.

He should let her go without the kid and tell her to not come back.