I'm not talking about knights, but men-at-arms and other proffesional soldiers. I'm aware
Who were they in the social strata of that time, depending on the country? How were they trained and financed? Were they sons of rich merchants and non noble land holders?
The most comprehensive study on a later medieval army in English is that of Adrian R. Bell, Anne Curry, Andy King, and David Simpkin, The Soldier in Later Medieval England, (Oxford, 2013). You can find out more about the project, and access their database on the companion website (http://www.medievalsoldier.org). I'd highly recommend picking it up considering your outlined question as there whole segments are dedicated to 'Men-at-Arms' (although the term is more troublesome than your clean distinction between knight/man-at-arms implies), 'Archers', and 'Where Did Soldiers Come From'.