Yes, especially if she's got the same surname as the kids. If the surnames are different, there is additional paperwork required to prove it isn't a kidnapping. I know someone who kept her married name for this reason.

But yes, it's very easy for a parent to move countries with a child leaving the other parent with bugger all legal options. While there is usually a requirement for a work/permanent VISA, the parent in question doesn't need to tell anyone about it, and the country issuing it isn't going to tell anyone if they are asked.

I say a VISA is usually required, there are situations where you can request asylum in a destination country. Everyone is immediately going to think of refugees in sinking boats trying to enter the EU or something like that. The problem the refugees in boats usually have is, they have passed through other countries. If you want refugee status, you have to ask in the first country you get to. (Example : If you were fleeing Liberia and boarded a plan that landed in Ghana and you then got onto another flight to the UK. The UK will tell you to piss off, because you should have asked Ghana)

There are cases where the opposite happens, family is leaving the country, so they go to the airport, parents board the plan and leave the kids behind. Yes, am aware that sounds made up, but it isn't and it's worse than you think because it was in a country with no social services nor any kind of facility that could assist.