Dismounted knights and men-at-arms weren't necessarily men who "preferred" to fight on foot. Rather, they fought mounted or dismounted as their commanders ordered based on tactical necessity. A man who had been ordered to fight dismounted might cut down his lance to be more handy on foot. Shortening the lance was common in the 14th century, but was starting to be phased out in favor of the poleaxe in the 15th century. The poleaxe was a very popular weapon for dismounted troops, owing to its versatility and effectiveness against armor. Swords and daggers would be near-universal for knights and men-at-arms as well. Unlike soldiers in video games like Total War, where units are differentiated by the kind of weapon they carry, there would usually not be separate units of heavy infantry for different varieties of hand weapon. Two men in plate armor would be deployed alongside each other even if one had a short lance and the other had a poleaxe.

EDIT: Also, I feel the need to clarify something in your question. The distinction between knights and men-at-arms was not military, but social. I will link to a comment by /u/thegreenreaper7 about the origins of the term "man-at-arms" and how it was used in contemporary sources. A knight did not become a man-at-arms if he dismounted to fight, nor did men-at-arms become automatically dubbed by getting on his horse. Contrary to popular belief, fighting from horseback was not necessarily the most prestigious role. At Agincourt, the cavalry wing of the French army was handpicked from the most experienced soldiers and the most prestigious aristocrats competed to position themselves in the front ranks of the dismounted vanguard.

some knights preferred to fight on foot, and formed units of men called men at arms

A knight and a man at arms are the same thing. A wealthier, professional soldier armoured and armed to the teeth. A knight was just a man at arms with the title of knighthood. It didn't matter whether they were mounted or not.