While going through basic training it was drilled into our heads that our guidon must never be taken by another unit, and must never touch the ground (if it was falling and you couldn't grab it you must "sacrifice" yourself for it). Now in movies you commonly see an individual/s carrying unit flags/colors/Roman aquila/etc, would those individuals carrying such items partake in any fighting or were they literally just supposed to stand there and guard their units standard, could they retreat early if it looked like a rout would happen, etc?
I'm going to answer for the 18th/19th-century British army, because that's what I have a source for (Richard Holmes' Redcoat: The British Soldier in the Age of Horse and Musket), but the answer is that colour-bearers could and did fight, and that it was an extremely dangerous position.
Colours could be taken to the rear in anticipation of dangerous action, and colour-bearers often died or received horrible wounds, because they were obvious targets and the enemy tried to capture colours: at Waterloo, one battalion had four successive officers and NCOs shot from under one of the colours, and another had their ensign sustain twenty-two sword wounds in defense of the colour.
In general, colour-bearers wouldn't lead an assault, but they would often be in the centre of the fighting as they were the focal point for their unit.