Not browband, not noseband. It's not a matter of "pulling through," just the whole bridle comes off.
With a snaffle bit, as you pull back on the reins, the bit pulls back in the mouth, which loosens the headstall on the face. You have the opposite effect with a curb, which actually tightens the headstall behind the ears when rein pressure is applied.
But if you have a throatlatch, it keeps the headstall from slipping over the ears.