You will probably win the fight to keep your daughter in CA and will not have to pay her fees. You want to protect yourself from your daughter being taken out of country without your permission. See US State Department website on passports on weblink.
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I'll respectfully disagree with the prior answer. There is a significant chance that you might be ordered to pay some PART of your ex's legal fees, unless her request is completely unreasonable and frivolous. (Why does she want to move?)
It doesn't sound like she's trying to "challenge" the existing agreement, but to modify it; custody orders are always modifiable, if the party asking for the modification can show the court that modifying the order is in the child's best interest.
Courts are required to "level the playing field" in considering requests for fees. It's unlikely that you'd be ordered to pay ALL her fees, but certainly possible that you might be ordered to pay some of them.
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