For speed in footwork linked to arm movments:
People who have to think about what they are doing are slower than people who do those same things only on their reactions.
Shadow box,get it exactly right with thought behind it so you move and do set ups for the right reasons. Study why you do certain things until you know why your doing them.
Only when you (KNOW deeply why) can you really think speed into and onto those same movements and combos.
Then do the same thing during shadow boxing and go take it to sparring without the thought process and you have freed up mindspace which relays over into your actions.
Specific trained reactive reflexes come from knowing not thinking.
Train train train until you know know know then stop thinking about it so your mind is free to analize the oppositon when your in action.
That is when real free speed is experienced.