100% correct. While this is not Islamic but in today's society she also have the right to poison the child against the father and cut ties between father and child. This is how society accept this now. Which fits very well the ahadeeth of the minor signs of qiyamah where children is close to their friends and distance from their fathers and another sign of qiyama cutting ties between family is rampant and that is usually child and father more than relatives cutting tie. I would say the rampant of father/child cutting tie easily hold 70% and more of the statistic where cutting ties in family vs two brother or sisters or uncle or aunt cutting ties. Although that is happening in my family here where we have ties cut between us and auncle and aunt.
If this question is about you, then yes you do have to pay child support and she does not pay a dime or should not pay a dime and it is obligatory on you to pay child support. It does not matter if she takes those child or children and leave the country and cut ties between you and them, you still have to pay child support. Just pretend you are paying bills and simply move on at that point if that happens to you. My recommendation is to adopt a child! Because with adopted child you as the man have 100% right on the child and no woman can take it from you, but your own child? That belongs to her, it is her property and she can do with them what she wants.