At one point I was obsessed with inductors and did a great deal of research. Here's a summary of that from my notes, edited together with some additional commentary.
The core issue before us, if you'll forgive the pun, is that (a) the iron core (iron slug, laminate, or ferrite) has hysteresis, i.e. it retains state between AC cycle and (b) the crossover point may slightly upwards shift with saturation.
The retained state performs a signal averaging between the old and new signals, i.e. distortion, as the flux is created/dissipated during the AC cycle. That distortion, furthermore, unfortunately is non-linear. How much does it in practice matter? Ahhh, well, without such arguments we'd be reduced to arguing whether or not Bon Jovi was better than Springsteen. (That's probably way too much of a Jersey joke. Better to make that a similarly contentious topic, such as whether Van Halen was better with David Lee Roth or Sammy Hagar.)
To understand hysteresis think of magnetic tape or wire recording where the magnetic domains retain state after the magnetizing current is removed. (Those old enough to remember floppy disks, or ferrite core memory for the geriatrics, will have yet another example.) The state may be retained for decades because energy is required to change that state. Which is what the AC current pumping through the speaker is: energy which changes state in inductors and capacitors. Now, the crossover inductor, of course, cannot retain state even a fraction of what is offered by the specially designed magnetic materials in tape. But the point is that energy is required to change the state of the inductor and that energy exists in the form of the AC signal moving through the speaker.
Like any other inductor, higher current levels builds higher flux, and as the flux increases the core moves towards saturation, i.e. the point where no more flux may be built. As core saturation increases the inductance decreases because the inductor is less and less able to absorb more energy, and thus increasingly behaves like a straight wire, passing the additional current unfiltered. (Chokes in a power supply are a common example of this phenomenon, where "critical inductance" specifies the minimum current required to keep the choke from saturating and similarly acting like a straight wire, having regulation fail.) As the inductance reduces, the crossover point upwards moves. (Smaller inductance higher crosses, larger inductance lower crosses. This is why woofer inductors are much larger than midrange ones, excepting, of course, the power demands of the woofer require somewhat thicker wire.) So the inductor begins to admit higher frequencies which the driver cannot properly reproduce and which overlap the driver following it in frequency. The higher-frequency signal does not damage the driver, so a woofer reproducing midrange and tweeter frequencies will not be damaged, but the sound will be distorted and overlap the actual midrange and tweeter further causing distortion.
While saturation is why the distortion worsens with power, the issue of crossover shifts and signal averaging always present to some extent, even at "lower" power because dynamic range offers fast-moving and high-energy peaks, aka headroom. (I am not addressing the significance, only the phenomenon.)
Lower frequencies have more energy and thus may more readily saturate the inductors. But higher frequencies requires fewer turns and may use thinner wire, so less reason to use cored inductors for that purpose.
But the issue and tradeoffs become more complex.
Not all iron materials saturate at the same rate. Mass matters, so a core with greater mass is less prone to saturation than one with a smaller mass. But the larger core weighs more and costs more. Laminated cores less readily saturate than do ferrite ones, which has implications for using ferrite for the woofer versus the midrange. But the fancy cores are more expensive to fabricate. What is essentially a transformer core constructed with EI layers may very differently perform than a stack of identical laminations simply together welded. Iron alloys differently perform than soft iron, so a laminated core of carefully tailored alloy has very different performance than a slug of cheap steel whose characteristics vary by the scrap metal being melted down. Slugs of ferrite are different than EI stacks, with the ferrite's saturation properties being more abrupt (akin to solid-state clipping) while laminations and stacks have a softer saturation (akin to tube clipping). Yet the inductance of laminations is frequency dependent, rising as frequency reduces and reducing as frequency increases. So while ferrite is better for higher frequencies and thus better suited for tweeters than would be stacks of steel, ferrite more readily and abruptly saturates at higher power than does a traditional metal core so the tweeter may have higher distortion at higher power. Eddy currents also exist in the cores, particularly at higher currents. Many times a cored inductor is itself kaizened not merely for performance, but to use less material because iron costs money and adds weight and the core may use poor quality steel which is less expensive.
On the other side of the equation, air core has more turns and thus higher DCR than iron core and, henry for henry, weighs significantly more. The additional DCR from an air core, BTW, tends to be negligible, particularly given that the nominal driver impedance varies across the range, but hobbyists often contentiously argue about this like David versus Sammy.
The ultimate point is that human hearing is often quite poor, particularly at lower frequencies, so the distortion from a iron-core inductor may not be audible compared to its air core cousin.
The main reason for manufacturers, even hobbyists, selecting inductors which are metal-core versus air-core, as has been correctly pointed out in prior posts, is simply money. An inductor is money in solid form. Since the iron age copper has cost more than iron, and the core substantially reduces the amount of copper, and thus the cost and weight, and a slug may be inexpensively used as a core, or even inserted into the finished inductor's plastic bobbin and then glued in place, instead of a fancy EI core. No more complex reasons exists. Same reason non-polar electrolytics are used instead of film. Just money.
Back in the early 1970s when the price of copper went through the roof, well, at least for that time, home wiring switched to aluminum and even aluminum inductors were used in speakers, with crimped-on copper pigtails for soldering. Bozak used such inductors, with ferrite cores, for a brief time. The difference in sound is noticeable. That may be a function of the higher DCR of the aluminum, poorly made copper-to-aluminum crimps, poor selection of the ferrite, or some other factor.
As with any of mass-produced speakers, simple changes — upgrading to air-core inductors and polypropylene capacitors, and lining the interior with cotton batting (not fiberglass) to reduce reflections, chamfering ports and driver holes, or installing plastic guides, etc. — can greatly improve the sound. Just look at the improvements to the Minimus 7 achievable for what amounts to the cost of a dinner out. (Well, in NYC that's probably the cost of eating a nice lunch out.)
We, as hobbyists, have highly affordable options for improving a pair of speakers which the manufacturer could not do and remain cost competitive against other speakers which did not have such designs. So the argument for us is different one, and small increments in performance for often larger increments in cost are often perfectly rational decisions to make.