... However, here is some info from the sozo website. Take it for what it's worth. ...
There's often a nugget of truth in everything; but often that truth is irrelevant in audio-frequency amps...
Technically; Several things happen. The major thing is; during the break in period, the dielectric material (the insulating material) interacts negatively with the signal flow. The dielectric absorbs and releases energy as opposed to passing it through the capacitor.
Dielectric absorption is real; however, it is a known characteristic of
different dielectric materials, which means some materials exhibit greater degrees of this characteristic than others. The "more perfect" the dielectric material (polystyrene, teflon, air, vacuum) the lower the dielectric absorption. I have no idea how long a cap has to be connected to an a.c. source to aborb electrons to its max capacity, but we generally have a decent d.c. source attached to coupling caps (albeit with a current-limiting plate load resistor). WIMA's cited test probably allows more charging current for the cap than a typical guitar amp, so maybe this factor is relevant to "break-in" though it also suggests that better dielectrics require less break-in.
There are other reasons like skin effect (With an alternating current, there is a delay in the magnetic field’s response to the change in current and the ‘old’ magnetic field tends to push the current towards the outside of the conductor. As the frequency increases, so does the effect until at very high frequencies the entire current flows in a very narrow skin on the conductor–hence the name).
Skin effect is real; however, it really applies at radio frequency and above. If you look at the chart under "Examples" on the linked page, you'll see the skin depth for copper @ 10kHz is ~0.5mm, but that's from each outside edge. If the wire is only 1mm think, skin effect has no effect. That said, RF stuff does have to be constructed with skin effect in mind, and microwave frequency equipment is so impacted by it that the conductors used are generally "waveguides" (see pic below). In those, electrical energy does travel through the hollow middle, but current per se is conducted along the solid outer shell. In any event, you'll probably face as much resistance-increase of your wiring due to poor prep/soldering as you do skin effect in audio frequency work.
One other consideration is self inductance (The property of self inductance is a particular form of electromagnetic induction. Self inductance is defined as the induction of a voltage in a current-carrying material when the current in the wire itself is changing). Although this is a non-inductively wound capacitor; meaning, careful attention is made to not produce inductance with the design, there is an inductance due to the alternating voltage.
This is poorly-written, because it appears to contradict itself. Self-inductance is real, and components like caps can be constructed to minimize their self-inductance. Again, self-inductance is most-relevant at RF because the apparent impedance for even a very small inductance gets bigger as frequency goes up. At RF, the self-inductance of a straight piece of wire might be significant; again RF circuits are constructed in a particular way to mitigate the negative effects. The result of self-inductance with a coupling cap would be some loss of high frequency output, but the added impedance will probably be exceedingly small at even high audio frequency and your speaker almost certainly rolls off massively more than you ever could through self-inductance in a coupling cap.
So dielectric absorption seems to be the biggest factor, though the posted information doesn't explain why any one brand of cap would need a longer break-in than any other cap using the same dielectric material (though I suppose a 10pF cap in a given material would have a shorter break-in than a 100uF cap of the same material, because there would be less dielectric material to soak up electrons).
