I'd also be interested to know if they were given uniforms, and what role family or place of origin played in how they dressed for the battlefield? Which was more important?

The concept of uniform is quite modern, you need the XVII century to have some homogenity in a unit, a concept that is fully implemented in the XVIIIth century. But in medieval times, you could have a sense of homogenity, if your captain/leader had issues some livery for his troops, for exemple, like during Charles the Bold times, or when the various commander in the war of the roses had equiped theirs troops (the arms of the commander where depicted on the livery). This livery was weared over the protection. You could also have some colour patch of other sign of recognition sewed on the soldier garment, like the cross of Saint-Georges during Henry V campain of France. During the Ghent revolt, in the late XIVth century, the revolted used white hoods as a sign of identifications.

Some sources: DEVRIES Kelly, Infantry Warfare in the Early Fourteenth Century : Discipline, Tactics and Technology, Woodbridge, Boydell Press, 1996.

JOLIVET Sophie, Pour soi vêtir honnêtement à la cour de monseigneur le duc de Bourgogne. Costume et dispositif vestimentaire à la cour de Philippe le Bon de 1430 à 1455, Thèse de doctorat en histoire, Université de Bourgogne, 2003.

MICHAEL Nicholas, Armies of Medieval Burgundy. 1364-1477, Londres, Osprey Publishing, 1983.