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DXY is a weighted geometric mean of 6 USD spot exchange rates which as you say is very heavily weighted to the USD:EUR rate. Futures and other derivatives, only affect it in the capacity that they affect the underlying spot FX rates. Given the size of the spot FX markets compared to the futures market I would expect that effect to be small.
I don't know about swaps on DXY but ICE (who calculate the DXY index) do list DX futures, which are futures on the DXY index.
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