Sorry to not be clear. What I am trying to say is that she set this ball in motion, not you. Her actions and choices put you in the position to decide to let her take the kids or not.
While it appears that it is your choice to fight her or let them go, that is a false dichtomy, she made the choice to leave you. Now she wants to have her cake - leaving her marriage and eat it, too - taking the kids with her.