Looting the dead (and injured, no Geneva convention) already happened while the battle was still underway, in quieter parts of the battlefield. Anyone in the area would start picking stuff, including (possibly recently homeless) local civilians (such as at Hastings). A battle would gradually transition from fighting to looting, whether on the field or at a siege (such as Bari 852). The objective of a commander at the later stage of a battle is thus to make sure that the main treasures (or the elite warriors) of the opposing force are attacked by the men he can control best, to make sure he gets the most out of the looting. The next part is either a retreat back to camp, which is usually rudimentarily fortified, or establishment of camp in a nearby fortified place, such as a town or monastery. If no camp exists (or unsufficiently fortified), a fortification is made quickly, particularly when in enemy territory (such as Dyle 891). The digging of mass graves for the dead is a job for the locals, in some cases the local subjugated population (such as Visby 1361). This job also started as soon as possible. Reshuffling of social status (partly on the basis of loot captured) could also take place immediately after battle, such as knightings or rewarding followers with silver or gold treasures or new ranks. The day's events would usually be concluded with a religious thanks to the saints and a prayer to the dead, though pagans (such as Vikings) might have taken a longer period in which captives would be sacrificed or enslaved. In any case, it might take a bit longer before a commander has sufficient grip over his army to continue campaigning (hence the fortification), for which purpose excessive (and quick) spending of loot on luxury products (food, women, alcohol etc.) would be a benefit. In the early medieval period, campaigns would be largely concluded after a decisive battle, so the army would split up in a looting spree through the countryside and then return to winter camps/home.

In traditional hoplite battles In Ancient Greece, controlling the battlefield after the battle meant you had won. The victor would put up a trophy and I believe they would often strip the dead enemy troops of their arms, which were very expensive back then. (The only way you could be a hoplite was if you had enough money to arm yourself. This was a critical aspects of class distinction in many cities in Ancient Greece, pre Alexander the Great)

This was in part due to the way hoplite armies fought each other. They were basically two massive lines that ran into each other until one side ran. If the battle was over and you were still armed and standing, it was because the other guys had ran away. The losing side would need to approach the victors and request permission from the victors to collect their dead. This was basically a formal acknowledgement of defeat.

Often times the entire war was decided by a single battle. In these type of battles burial rights were almost always given to the losing side. Proper burial was extremely important to all Greeks. Though during extremely brutal and 'uncivilized' wars like the peloponesian war, some burials did not take place. This would be extremely demoralizing to the defeated city. Famously in that war, Athens executed many of their generals at a critical point because they were unable to collect their dead due to a storm after a large naval battle with the Spartans

Reference: Thucydides - The History of the peloponesian War

Donal Kagan - Yale open class http://oyc.yale.edu/classics/clcv-205#sessions (^really good)