During the Hundred Years War era, noblemen captured in a battle were normally held for ransom. In such a case, who got to keep the prisoner, provide him lodgings, and receive the ransom? Was it the soldiers who captured him? The lord which the soldiers served? Or the sovereign which the lord served?
Noble prisoners captured by other nobles would be held by them - in the proper manner- which frequently got so long and expensive the nobleman lost out on the deal.
Since ultimately they belonged to the king, who in the 15th Century would be leading the army in the field, the noble would have to pay a fee to the king to keep them - essentially a cut. Or they would sell the prisoner on up the chain either for a cut of the ransom or for advancement.
Because of the nature of warfare in this era a commoner was unlikely to capture a high ranking noble, knights sought out and fought other knights. Agincourt is an exception in that most of the French nobility were slaughtered in the mud by common soldiers and archers.
There are a bunch of interesting brothers-in-arms contracts from the 100years war where people agreed beforehand to partner in capturing prisoners and guaranteed to pay a certain part of any ransom for each other. They read almost like car insurance deals.
A prisoner captured in battle belongs to the "state" of the soldiers who captured him. In theory, that would be the king of England or France.
Now it is possible that with at least some high ranking nobles, the king of England or France would let the capturer keep the prisoner. But this would be a form of "delegation," not a usual practice.
But common soldiers wouldn't keep the prisoner. They'd hand him up the "chain of command" to an officer, their lord, and ultimately the king.