After you beat the elite four, wally is in the victory road cave, just inside exit. You can still battle him there from time to time.
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No you can't use the azure flute after you beat the elite four
You wait until you beat the elite four and professor Elm will give it to you when you beat the elite four.
You then have to beat Cynthia
you get him after you beat the Elite Four 1,000 times. Trust me, it took me 1 year to beat the Elite Four just 500 times.
You simply cannot.