The idea of a woman taking a nursing infant along to meet a man in another country is a crazy idea. Is there any way to change the woman's mind? Maybe get a lot of family members together to go and try to talk her out of it? There's just so much that is wrong about the situation. Maybe too, your sibling can get rid of internet service for awhile too. 

Given that they already have two children, he should do what he can to keep this family together for the sake of his children if nothing else. If he wants this family of his to work out, he needs to figure out what her motivations are and why she wants to go.

What's this travelling to another country business? Are we talking Detroit to Toronto? England to Scotland? Or the US to another continent?

America to Australia 

Maybe he should ask her to marry him. Never to late to correct the situation. maybe that's what she's waiting for. 

This is OP.  He already asked her to marry him and she said no.  He figures the relationship is over for good.  That is probably for the best, but we still need ways to limit the damage to the children, as much as possible. 

Given that they already have two children, he should do what he can to keep this family together for the sake of his children if nothing else. If he wants this family of his to work out, he needs to figure out what her motivations are and why she wants to go.

What "family"?  This is an illicit union, and the two shouldn't be together at all.  Man and woman here CANNOT stay together, living in sin ... unless they get married ASAP.  Unfortunately, the two fornictaors (adulterers?) brought children into the world, but based on this man's morals or lack thereof, what does it even matter who gets custody?

But, in short, they're required to separate anyway due to the sinful nature of this cohabitation.  Under some very strict circuмstances, the Church will allow a couple with children to continue living together but ONLY if there can be assurances that they'd carry on as brother and sister and that there would be no scandal.

But the end cannot justify the means, and the "good of the children" cannot justify continuing to live in a state of mortal sin.

So it's completely moot to speak of keeping some "family" together when they are required to separate due to God's moral law.

And what does an "affair" mean?  Since they're not married, it's not as if this woman would be violating any rights a true husband would have to fidelity.  She's ALREADY in an affair with the current man.

I have a close relative (e.g. sibling) in a common law relationship with a woman with whom he has two children. His "wife" has developed an online friendship with another man in a foreign country.  She is planning to visit this other man (whom she has never met in person) for a month. She is planning to take the younger child, who is still a nursing infant, along.  It is hard to believe this is merely a platonic online friendship.

Your brother has no right to anything other than a "merely platonic ... friendship" with this woman either, no right to any kind of "fidelity".