Hi, this answer's coming a little late considering when you'd posted, but it's always best to get one late than never, isn't it?

To start; while it is correct that troops and armies would be drawn up from the local population in the event of a war, this was only so far as individual companies beneath the local lord's banner; the troops fighting for the duke of York, for example, would likely have been from the towns and villages under the duke's direct control. In terms of more general army composition, however, we'd have to consider that this would likely be a great many dukes, earls, barons and knights from different parts of the country all coming together to fight in a campaign.

To continue with an area I apparently am a little too familiar with, the army mustered by Henry V of England for his Agincourt campaign was one drawn up from almost the entire country, except only the Welsh and Scottish marches, which were actually reinforced to avoid foreign powers on the British Isles from taking advantage of the English army's invasion of France. The Agincourt campaign, it is worth noting, was one of the more larger scale operations of the war, so it goes without saying that troop count would've been much higher and widespread than others.

Service for the campaign was required from most English counties, as well as 'in our duchy of Guyenne [in southern France], or in our realm of France', so troops were drawn up from the fullest extent of the English kingdom, and brought to Southampton in preparation for invasion of the French mainland. We also have the duke of Clarence, the king's brother Thomas of Lancaster, who drew up 240 men-at-arms and 720 mounted archers from his own lands in Normandy; his brother Humphrey, duke of Gloucester, who drew up 200 men-at-arms and 600 mounted archers from his lands in England; as well as Edward, duke of York, Thomas, earl of Dorset, and Thomas, earl of Arundel, who each fielded 100 men-at-arms and 300 mounted archers from their own lands. England also fielded some rather impressive specialist archer companies from Wales, Cheshire and Lancashire.

France, equally, had troops from virtually all over their kingdom joining the ranks to fight against Henry V; men from Burgundy, Savoy, Aquitaine, Normandy, Brittany, Brabant, and many places besides. The only regions of France that received only light recruitment in 1415 were those south of the Loire, 'save for the companies of Bourbon and Orléans'; surprising, really, considering it was in these lands that the dauphin possessed a rather large and specialist company that went unfielded during the campaign.

While these are only two examples, from the same campaign even, they are indicative of wider practice during this time; during the Wars of the Roses that followed the Hundred Years' War, for example, you might find companies even from northern France fighting in battles in England. The Battle of Bosworth, which ended the Wars of the Roses in 1485, saw troops from Wales, Normandy and Brittany (albeit largely mercenaries) fighting under Henry Tudor's banner, while Richard III's troops had been mustered largely from the southern counties before gathering at Leicester. I wish I could give more thorough and wider examples of how this sort of recruitment worked, but I think the Agincourt examples work rather nicely as smaller parts of a wider trend; while troops themselves may have been recruited locally by their lord, they would have gone on to serve in a larger campaign force built up of numerous companies from across the kingdom, all of whom will have been recruited by their local lord in a similar fashion.

I hope this answers your question, feel free to badger me with more if it doesn't, I'll try to help if I can!

Sources:

Juliet Barker, 2015, Agincourt: The King, The Campaign, The Battle

Anne Curry, 2002, The Hundred Years' War: 1337 - 1453

Anne Curry, 2015, Great Battles: Agincourt