I'm referring to late medieval knights 14th century or so.
First of all, the helm. Most suits of armor shown in museums and by medias have closed helmets or mobile visors that severely restrict vision. This is all well and good while on horseback or during a tourney, but did they really wear those while fighting on foot too? I'd say situational awareness and not being jumped from behind are more important while in melee, but I couldn't find informations about this. I imagine that, having squires and in general an entourage of people to support them, they could have more than one helmet at their disposal, but again, I struggle to find reliable informations about this fact.
Also, I know that tournament armor and battle armor were different, with the latter being lighter and more mobile, but did they also have different armor for when they were to fight on foot rather than on horse (if they knew in advance of course)?
Full plate harnesses didn’t actually become the norm for professional warriors until the early to mid 15th century.
That being said, there is currently no consensus agreement among historians on how knights and men at arms preferred to wear their helmets in close combat. Most likely it came down to each man’s personal preference and how comfortable he felt in his armor. If you look around you can find contemporary depictions of men fighting with closed face helmets, open visors, coifs only and a variety of other combinations. When it comes to armor, you are always trying to strike a balance between protection, mobility and awareness.
Like you mentioned, wearing a closed face helmet like a Hounskull Bascinet or an earlier Greathelm would certainly provide a high degree of protection for the face, but severely limit your vision, breathing and hearing.
In my personal opinion, I think the most common (but not universal) thing to do would be to keep your visor closed while on horseback or while advancing under missile attack, but then open it or remove it when melee was engaged.
I’m sure this post will get removed since I didn’t include any links, but figured I’d throw in my 2c.