It seems like there are many depictions of fantasy characters wearing their cloaks while being able to maneuver well in combat. I am curious if there is any truth to this. It seems like it would hinder or put the warrior at a disadvantage. (I ask about the Spartans mainly because they are the first cloaked warriors that I can think of).
Trajan's column appears to show some Dacians wearing cloaks in battle, here. That seems to me to be as close to a primary source as you're going to get on someone wearing a cloak in battle, as many typical battle narratives might well not mention the precise gear of the soldiers, and lists of soldiering gear might not mention if cloaks are for battle use or just for less active parts of soldiering (guard duty and the like, where you could presumably lose the cloak before running into the fight).
I'm making a few assumptions here because this isn't my specialisation: first, that this is a battle (based on the shields the figures at the bottom right have) and not some sort of massacre, and second that those are Dacians (as I understand it, wars with Dacians are the main focus of Trajan's column, and these figures certainly seem to fit some barbarian stereotypes).
I seem to recall that polybius in his description of a battle with gaesatae stating that those who went into battle naked suffered a lot of pain to projectiles that got past their shield while those wearing thick cloaks were protected against glancing blows.