Yes, agreed, they would most likely be bandits. I still don't think it's completely out of the question people like them would just stroll in a village pub and ordered a beer and sausages but it's definitely very unlikely, yeah.
The first thing I picture a group of deserters would do to get food in an unfamiliar area is they would drop by some isolated farm at night time and steal a pig or a sheep. I don't see them entering villages openly for the same reason Henry wouldn't go into a village that he recently has been stealing from.
Discipline was actually no joke in late Medieval armies, or at least not as much as people imagine. One of the most comprehensive sources of information on late medieval army organisation is Charles the Bold's edicts ("ordonnances") written between the late 1460s-1473, which prescribe the equipment and organisation of his army. Since they summarize the experience gathered from the Italian campaigns of the mid-to-late 14th century, things didn't differ much in Italy. The main unit was the company, which numbered around 900 men. The smallest unit was the "lance" (9 men), six lances make one "chambre" (54 men), four "chambres" make a "squadron" (54*4 = 216 combatants), and four "squadrons" of 225 men each make a company (total 900). Every unit's commander would keep track of how many men he has and would know them.
The ordonnance deals with how discipline must be maintained and the company commander (the "conducteur", obviously from Italian "condottiero") and his subordinated squadron commanders ("chefs d'escadre") are charged with assuring the proper behavior of their men towards the population. They were obliged to appoint a special commissary to record complaints from civilians and reimburse any damage, deducting the money from the guilty soldier's pay. During march, going out of formation to pillage or forage on your own accord was forbidden. You could lose your wages, have your equipment confiscated, or if in enemy territory, sentenced to death. Desertion was a very serious offence, and if someone was reported to have deserted, marshals would be dispatched to await him at his place of residence to arrest him, which tells you that soldiers would report their place of residence when enlisting in the lance. See "Men-at-Arms #144 Armies of Medieval Burgundy". There is a nice big pack of MaA magazines on piratebay.
I'm not saying it would have been that strict in Sigismund's army, but if you were a deserter you would definitely try to stay out of sight and not just welcome strangers walking into your camp. They might turn your location in to some foraging party of your former army, and they would deal with you to make an example. You were on the run first and foremost if you were a deserter, that seems most likely to be the case. They would hide in the woods and try to make their way back out of the theater of operations and into Hungary with minimal risk.
It is generally accepted that prior to Thirty Years' War, the level of destruction you speak of was very rare
The Italian wars (of the 14th and then late 15th centuries) weren't all that destructive, but the Hundred Years' War was pretty bad on civilians. In Italy, they would free prisoners after battle and would only keep ransom-able nobles as prisoners. But towards the end of the 15th century, when German and Swiss mercenaries became more frequent, things got rough. Germans and Swiss were notorious for killing prisoners, whether in order to loot their belongings or just to save on provisions. Booty was considered as an alternative form of payment for the mercenaries, so the sacking of villages and mass killing of prisoners was a way to get a bonus or skip this month's payment. Payment was usually kept in arrears of at least 2-3 months, to dissuade potential deserters, who would get their payment and wander off. This practice also encouraged soldiers to turn to looting, because they were expected to pay for their provisions out of their own pocket.
I think one difference in the medieval mindset we tend to underestimate is the tolerance for witnessing and incurring violence that people had. For a medieval man or woman, witnessing the death of peers, relatives and siblings was far more common. The average Western person today would be disturbed by the sight of blood or from witnessing someone hitting an animal. Most of us here likely don't know how to efficiently cut the head of a chicken
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People back then were much more used to such sights, but mass slaughter of prisoners or the razing of settlements would still make a strong impression, especially on a young person.
There is no preaching of "sheepish tolerance". You can kill these Cumans and even if you befriend this particular group you can slaughter another group of Cumans the "friendly" ones warn you about. You are also showing that you haven't really understood the plot. Cumans aren't "invaders". They are mercenaries brought by Sigismund to support an already existing rebellion of Czech nobles against his brother Wenceslaus. They are invaders in same sense that Hessian mercenaries hired by the British were invaders in the American Revolution.
I already mentioned you're allowed to kill them. My problem is with the way the game is presenting them, and the way that NPCs don't miss mentioning they are nice folks whenever Henry asks someone "Do you know about any Cumans in the area". The game is obviously pushing a viewpoint on the player and it's quite an implausible one, in my humble opinion.
I understand the plot quite well, and the Cumans are invaders, by virtue of them being soldiers in an invading army, or you would disagree?
Also, Cumans originally settled in Hungary in 13th century fleeing from Mongolian conquest. I suppose 13th century Hungary was also woke considering it "sheepishly" allowed such numbers of refugees to settle there.
I don't have any sources at hand to confirm or deny that 13th c. Hungary was woke. But I'll look into it in whatever books I have at home.