At the end of a battle I would assumed there would be a ton of damaged leather or metal (type of metal depends on age and area) on the battlefield. Would the winning armies just leave the armor to rust/rot on the field or take the leather and make it into other items and melt down the metal. I would imagine that metal was too expensive to just leave laying around so they must have taken it with them. Also broke weapons could possibly be reforged?
In terms of intact enemy weapons and armor what would become of them? I know that many armies did not have standardized equipments so I could see them just appropriating the equipment but armies like the Roman armies might not (specially not weapons)? Would they give the weapons/armor to their auxiliaries?
It looks much like you'd expect: there was a lot of opportunistic looting after a battle.
Sometimes, if the armor was old or broken, and the victorious army didn't want to lug it home to recycle the materials, it might be left on the bodies. You see this at the graveyard from the Battle of Visby (1361), whose archaeological excavation found a number of bodies buried in old, outdated armor that aparently wasn't worth saving.
In some cases, the valuable pieces were chopped off the dead and carried away; the recently discovered Staffordshire hoard (7th century) contains bits of gold that were broken off of swords, a helmet, and other things most likely looted from a battlefield, carried off to a crossroads, and buried for later recovery (this is the most recent interpretation, at least).
And of course, the winners could upgrade their own equipment from the dead; the 9th century poem Beowulf talks about looting cweapons from corpses, and later Scandinavian laws (11th century) tried to make this illegal (which may suggest it happened a lot).
Broken equipment could always be recycled or repaired. Leather straps wouldn't be much good on their own, but iron is easy to reforge (and there's archaeological evidence of broken weapons and tools being recycled), and broken armor could be repaired.