They were foot archers (a few of them foot crossbowmen) travelling on horseback and at some point during the 15th century or very early 16th century, the captains of their companies decided on occasion to equip them with demi-lances and use them and their horses as mounted scouts, skirmishers… Quite logical.
As for their evolution and possible roles on the battlefield during the 16th century, I can’t say for the moment. I’d need to do more research.
Their evolution from foot archers to lighter heavy cavalry seems to happen through the first quarter of the 16th century, and is not easy to document. At Fornovo (1495) they are still foot archers of the Hundred Years War type, with horses for campaigning but generally not expected to fight mounted. For an offensive campaign abroad, like the 1494 invasion of Italy, some crossbow armed 'adventurers' might have been hired to fill out the ranks. The role of second-class light armored lancers is filled by coustilliers, who are closely integrated with the gendarmes they support.
While not intended as heavy cavalry, Archers of this era were experienced professional soldiers, often with decent armor, and could fight on horseback when they needed to. Most archers at Fornovo fought dismounted, but the Royal Scottish Archers, who were an elite bodyguard unit of this type, remained mounted and helped withstand a charge by Italian gendarmes. There is not much evidence for them shooting from horseback, as the John Tiller game seems to imply.
By Ceresole, in 1544, their main role is as lighter lance-armed cavalry supporting the gendarmes (much like coustilliers had been earlier), or sometimes detached to deploy on the flanks. They differ from cheveau-leger primarily in being recruited and paid through the regular gendarme companies, rather than recruited
ad hoc from volunteers or local allies.
The evolution happens sometime in between, as the French become increasingly reliant on Swiss mercenaries for foot troops, then begin to develop their own permanent pike and shot formations. It would be great if we had some better documentation of what they were doing in the battles of the 1510s and 1520s, since this is likely when a lot of the change was happening.
So, they were indeed ordinary archers until the 16th century, and then gradually became lightly armed cavalry, am I right? Did they further evolve into fully heavily armed knights later on?
By the middle of the 16th century, 'archers' were heavy cavalry, probably still lighter than full gendarmes on average, but not by much. They fought in the gendarme units and came from the same social classes - a lot of archers were younger soldiers, essentially waiting their turn for a spot as a gendarme to open up (since the crown would only commit to paying a certain number of men at the gendarme level). After the heavy casualties during the civil wars of the early 1560s, a lot of 'archers' were promoted to gendarmes, and I don't believe this required much changing of equipment.