Well, the main thing is that they generally didn't.

The only prisoners taken were typically of high birth, nobles and aristocrats. If you were a dirty mercenary with a spear you were likely to be slaughtered or just sent on your way if the enemy was feeling merciful - usually without your weapons, armor, and money.

There were no large scale prison camps or prisons at all - highborn captives were usually held in pretty decent accommodations, often a tower or nice house until ransom could be arranged. Ransom was usually done through intermediaries sending messages back and forth and could often take awhile. Sometimes they were even released "on their honor" to go raise their own ransom and return with bags of coin to pay their captors.

The logistical nightmare you imagine does not exist because at no time would anyone want to shoulder the burden of medical treatment, food, and housing for masses of useless mouths.

What game are you modding? If you were aiming for historical accuracy then you can ignore most of it. There were no POW camps full of soldiers writing letters home like we see in modern wars. Prisoners were slaughtered, ransomed, released (usually robbed, sometimes mutilated), or recruited into the victor's force (many soldiers at this time were simply mercenaries with no real loyalty to anything but money and plunder).

Typically the victorious army after a campaign would dissolve as most of the men are mercenaries moving onto the next fight or lords pledging their retinues for a season as they need to get back to their lands to oversee them. Large permanent military bodies were extremely rare at the time.