What happens to Mjolnir (Thor's hammer) at the end of Endgame?

Cap probably brings it back to the past.

I can't provide a picture, but Mjolnir is seen at his side when he goes back to place the Stones in the past, and it's nowhere to be seen when he gets back. I think he knew that thing had to be given back to time, to preserve the original "continuity", and did so.

As to when and where to return, it's not that hard to figure out: ask Thor when he picked it up, go to that very moment, sneak anywhere on Asgard, drop Mjolnir on the ground, go back to Peggy/the rest of the Infinity Stones. Past Thor can summon Mjolnir at will, and probably isn't that much regardant of where he left it - the hammer will just zoom into his hands.

Steve takes Mjolnir back into the past with him, you see it on the floor of the platform. Presumably he takes it back to Asgard, but, he could equally well as keep it for himself... we simply don't know what he does with it.

If he does bring it back, how come he knows the exact time and place to return?

It is safe to assume either Thor or Rocket told him the time this happened.

Well... have you wondered why did it take a while for Mjölnir to reach Thor when it was summoned? Maybe once when it took longer than usual it was because Cap placed it somewhere a little far away when he jumped back in time one last time....

And most importantly, Banner made it clear that your past doesn't and can't change the future in which you are... although that was contradicted when Cap showed up again as an old man... that too with his shield, all shining and new.

When Thor "borrowed" Meow-meow (that's its proper name) from 2013 Asgard, he had to get it back somehow... otherwise Thor 2013 would have noticed it missing, and found himself with no hammer at all for The Dark World, the first two Avengers movies, etc.

So it had to be returned to 2013.

Since Cap was going to 2013 Asgard anyway to return an infinity stone, he simply returned Mjölnir as well. That's why he took it with him at the end of Endgame.

The result of this is that Thor experienced the normal timeline as depicted in the films, but Meow-Meow had an alternate experience in the middle of The Dark World where it went to 2023 and fought 2014 Thanos, and then came back to The Dark World and finished the movie. If only hammers could talk...

Mind you, Thor didn't need to deliver to a particular place, only a particular time. It can find Thor.

Cap goes back to the forties to replace the first stone, then lives a quiet life with his girl, replacing the stones as time goes on. This doesn't create a new time line because he doesn't change anything. He has to go totally unrecognized for this to work.

So what about Mjolnir? This means that from the 40s up to Dark World, there are 2 Mjolnirs, and somehow Thor doesn't notice. Maybe only the closest one comes when he summons it.