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Do you have any AAdvantage miles to redeem? Although US will join oneworld at the end of March, the Dividend Miles program is not going away that soon.
And the issue is not whether CX, MH, and/or QF impose surcharges on award flights; it's whether the FFP whose miles you are redeeming assesses surcharges for redemptions on CX, MH, and/or QF. Currently, if you redeem AA miles on those carriers, no surcharges; if you redeem BA Avios on those same carriers for long-haul travel, you'll be assessed hefty surcharges.