Family right

Well, first, I'm so sorry to write here in english, I do not master swedish well enough, but I'll learning.

Here is the situation : I french and my girlfriend is swedish. She has 3 kids, and here ex told her than if she leaves him, he would take the kids away from here. She has a depression, and issue with alcohol, but she is working hard to quit, even he doesn't help. Anyway, my question is, I don't know how things work here, but in France, there is no way a mother be forbidden to see her kids, actually, she would almost always have the custody, except if it has been proven she might be a danger for them (if they have been beaten, physically or emotionnaly).

So...is he just trying to manipulate her to keep her with him, or does he have the right to prevent her to see them again ?

Thank you for your time