While studying the 100 years war, I noticed that after Agincourt, when the English were extended into France, there seemed to be no recruitment of French people into the army. This confuses me for to reasons:
I am unsure what recruitment was like during this time, whether it was still feudal levies or something different. If the system was the still the feudal system then surely the French would have been used to providing a small number of men.
Gascony and Aquitaine had been in English hands for a very long time by now. Surely their loyalties were so in line with that of England that soldiers could have been gathered from these regions?
Am I incorrect about these presumptions? Did the English, in fact, recruit French people into the military? If not, why not?
I'm not qualified to comment on the first part of your question, but numerous Gascons fought for the King of England (who was also the Duke of Gascony) throughout the Hundred Years War. They may not have been present in large numbers on every campaign, but this had to do with the need to maintain garrisons in Gascony itself. The Poitiers Campaign of 1356 is a major exception. Edward the Black Prince led a mixed army of English and Gascons north from Bordeaux as far as Tours, before being cut off and forced to give battle by the French (bad idea).
As a side note, I think you may be falling into the trap of assuming that medieval kingdoms corresponded to modern nations. The Hundred Years War helped to form English and British nationalism, but that nationalism took hundreds of years to develop. For the most part, it did not exist in the 14th-15th centuries.