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If there is no custody order in place from the court preventing a move, then there is nothing that would keep her from being able to legally move with the children. If they are married, then if both children were born during the marriage, then the presumption is that he is the father, whether he is on the birth certificate or not.
The only way he could stop her from moving would be to file for divorce and then seek an emergency injunction preventing her from moving pending the outcome of a court ordered DNA test. If he is determined to be the bio father of one or both of the children, then he can get a custody order from the court giving him custodial time.
But if she were to move without giving him any notice, then it would be hard for him to get the injunction since he wouldn't be aware of the move. He would then have to file the divorce and paternity action to get a custodial order (if he is the bio dad) to establish any custodial rights.
However, in order to prevent him filing a complaint for "parental kidnapping" she would want to notify him of where she is living once she moves so he can't say that she is hiding his children from him.
She doesn't have to take any actions, she can just move. There is nothing preventing her from doing so since they both have exactly equal custodial rights to the children. So whoever has them has the legal right to do so. However, until she lives in Nebraska for 12 months to establish residency, if husband filed for divorce and custody, she would have to return to CA for any court hearings because that would still be her legal state of residency until she hits that 12 month limit.
If he didn't file for divorce and custody within 12 months of her moving, then she could file in NE and fix the jurisdiction there for the case and he would have to travel there to any court hearings.
But CA has much higher child support awards than NE, so it would be in his favor if he waited to allow her to file in NE as any child support payments would be much lower.