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What EU/Schengen rights does my wife (US citizen) have once she gets her non-EEA family member UK residence card? Is she allowed to travel freely within the EU? How long can she stay in individual countries? Does she still have a 3 month limit for all of Schengen? Would she need to apply for a local residence card after a while if we were to move to another member state?

Thanks!

If entering another Member State on her own (not to join you), her rights are equal to any other US citizen.

If travelling with or to join you, your wife's rights equal yours.