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Ime preserving dynamics has to do with minimizing those things that cause compression, namely voice coil heating, subsequent magnet heating, flux modulation, and mechanical system non-linearities.
Voice coil heating causes rapid-onset thermal modulation due to the rise in resistance as the voice coil heats up, and then the voice coil heats up the adjacent magnet, and the magnet loses strength until it cools back down. The magnetic flux in the voice coil also modulates the strength of the permanent magnetic field a bit as the two interact. And suspension systems can go non-linear at high excursion, which can also drive the voice coil outside its linear range.
Woofers and tweeters were mentioned in the original post - it is not uncommon for woofers and tweeters (and mids) to have different power compression characteristics, resulting in a change in tonal balance as the speaker gets pushed to higher sound pressure levels.
One solution is to use drivers that will be operated well below their rated thermal power handling even on peaks. For instance Klipschorns are theoretically capable of something like 125 dB, so even on 112 dB peaks they are only seeing 1/20th of their rated power.
Another key, more practical, and sometimes overlooked, design aspect when it comes to dynamics are the crossovers. Passive crossovers typically rob considerable amp power - most especially in the dynamic peaks which can be rather blunted and compressed. Removing passive crossovers and going active can be an ear-opening experience. Another option would be to either look at speakers that are known to employ rather high quality passive parts (perhaps the majority actually don't), or consider investigating how to upgrade with better quality parts.
besides what Duke (audiokinesis) wrote about the importance of not putting the drivers into mechanical compression, I think that ivan_nosnibor brings up an important point - cross-overs.
For me the most dynamic speakers are time-coherent speakers that use a 1st-order internal x-over. Not saying that an external x-over would have been better but I’ve not heard a time-coherent speaker w/ an external x-over so I cannot comment.
Assuming that the speaker designer chooses the drivers correctly so as to not induce mechanical compression, dynamics in speaker comes from its actual design....
Thermal compression is just one non-linearity in moving coil loudspeakers.
Diaphragm misbehavior (due to high stress at high drive) in another factor.
The most common factors are suspension non-linearity (as the diaphragm moves) and magnetic non-linearity (as the coil experiences a varying magnetic flux density as it moves).
Using a very large loudspeaker (one actually developed for much larger spaces) in a smaller space offers many advantages.
For me, the only significant weaknesses in an all horn system used in the home are frequency response smoothness and time and cavity issues. In the Klipschorn, there is very significant time and cavity issues in the upper end of the bass horn as it mates with the low end of the midrange horn.
I heard that the most recent versions of Klipschorn are much improved in this regard but still, the bass source is much delayed compared to mids and treble.
That (and cost) might be one reason most horn systems use more conventional direct radiating bass approaches.