The one you need to find mesprit is called the marking map you can get one from a guy in the pokech building in jubilife city (he has brown hair ).
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to get a lv.1 mesprit you need to intter an cheat code.
U NEED POKETCH APP MESPRIT IS STILL IN CAVE YEYZ
Its not Azelf its Mesprit that flees and you get it by using the Pokemon tracker app and you wil need a Pokemon with mean look so Mesprit does not flee.
there is nothing you really need to do with azelf, uxie, and mersprit. you just need to see them to complete your pokedex.
you need azelf,uxie,mesprit to open the cave after beating the Pokemon league