High dynamics!

also headroom in my experience, if you drive a system to it's limits, it start to compress, that happens when a speaker goes into "compression" when the cone is driven hard and floats dynamically (moves somewhat uncontrolled) as it produces sound. Amp compression mostly happens when the PSU or the output transformer (for tube amps) saturates when it's pushed to it's limits. Driving speakers and amps way below their limits let the music uncompressed, so with full dynamics present.
 
That too; normally it goes hand in hand with efficiency, but if the available linear travel is small, there will be an increasing likelihood of compression at (usually) the lower end of the BW. Unfortunately, in practice Xmax tells you relatively little, so in an ideal world you'll know the physical limits & measure the distortion performance HD2 - HD5 at a variety of drive levels.
 
Right, high dynamics = headroom/peak SPL at inaudible distortion, so the sum of Eff., Xmax gain with > ~2%/95 dB for 'FR' driver choice in an early forum ~consensus, though with so much based on THX nowadays, ideally need ~99-102 dB eff. drivers if low Xmax to meet its 105 dB min. transient peaks depending on how Xmax was derived and how low the speaker is tuned.
 
What creates high dynamics in an element?

Mass, or lack thereof in the moving part. I'd say it doesnt so much create, as "allow"...

Maybe boxes make a difference in the dynamics?

Seeing as you want to hold the driver in a fixed position as still as practically possible, I'd have to say they do. Looking at some designs, the driver is not only gripped or held in place by its flange, the internal bracing structure puts mechanical pressure against the magnet pole piece as well. I'd think such a design would make a difference in the dynamics of the sound.

I cant begin to fathom the tolerances involved, say, when you screw down the driver to the cabinet baffle, it simultaneously puts pressure on the back of the pole piece, via a brace fixed to the inside of the cabinet walls. Just enough force pressure to make a difference in the dynamics, but not so much as to warp the driver basket... I suppose with CNC fabrication machinery - or a lot of carefully executed fitting work without.