After you catch the main legendary (dialga, palkia, giratina) go to lake verity and mesprit will be in the cave in the middle of the lake. Talk to it and then it will flee and appear on your map app.
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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 ).
When a Pokemon is travelling around (mesprit, cresselia) the marking map will show a little face. Wherever the face is on the map, you will find the Pokemon. The Pokemon wil not be standing there, it will be hiding in the tall grass or even on water..
In map 22 (Ground)
Mesprit moves around the world at random so it could potentially be anywhere after it runs away. To find where he is, use the map on the Poketch.
To find Mesprit after it runs away Note:Happens after beat Galactic at Lake Valor aslo when you beat the elite four and got the national dex go to canaclave and go to the saliors house talk to the boy and he is having a neverending nightmare from darkrai talk to the sailor and you can go to fullmoon island and talk to cressila and will run of and track it on the marking map like mesprit.