The Red Cross was founded by 1859 over these circumstances:
Until the middle of the 19th century, there were no organized and/or well-established army nursing systems for casualties and no safe and protected institutions to accommodate and treat those who were wounded on the battlefield.
(...) Jean-Henri Dunant, in June 1859, (...) arrived in the small town of Solferino on the evening of 24 June after the Battle of Solferino, an engagement in the Austro-Sardinian War. In a single day, about 40,000 soldiers on both sides died or were left wounded on the field. Jean-Henri Dunant was shocked by the terrible aftermath of the battle, the suffering of the wounded soldiers, and the near-total lack of medical attendance and basic care. He completely abandoned the original intent of his trip and for several days he devoted himself to helping with the treatment and care for the wounded. He took point in organizing an overwhelming level of relief assistance with the local villagers to aid without discrimination.
Also, I have read useful questions (and its answers) such as How did ancient armies keep the route of supplies to their battlefield?.
Still, I wonder: what was the usual aftermath of a battle during the middle ages? How would wounded soldiers proceed afterwards? Was there any group of rivals executing the survivors or were they just left behind? Was part of citizen's duties to go and 'clean' the battlefield once the battle was over?