Is the father entitled to travel with the children to another country to visit them, given that their custody belongs to the mother, and is the mother entitled to refuse for fear that they will not be returned to her?

If custody belongs to the mother, and the father wishes to travel with the children for a visit, not for relocation and permanent departure, then this does not forfeit the mother's right to their custody. The right remains hers, and the father is not permitted to travel with them without her permission.