Drive units do change a very small amount in the first few moments of operation, I demonstrate this.

The What HiFi interview get's that part right, but they inflate the effect by an order of magnitude or two.
The effect of run-in is absolutely
tiny, on the order or even less than environmental effects like changes in room temperature. The modification of Qms due to changes in spider deformation for instance are similar in magnitude to the mechanical force of my house ventilation fan blowing air over the surface of a driver from 2 meters away, on low. Putting a driver in a refrigerator for 5 mins has a larger effect than break-in. Operating a driver at 1 Watt for a few minutes causes much more change in a driver's mechanical and electrical characteristics than any run-in phenomena. These changes due to run-in result in response changes that are smaller than moving the finished speaker a few cm relative to room boundaries. None of these changes with run-in are significant enough to change the alignment of the box or application. I have demonstrated this. Vance Dickason describes the complete nothing-burger that is driver break-in in his books on speakers . Andrew Jones has discussed the run-in phenomena, the physical mechanisms behind it, and how absolutely tiny it is as well.
The idea that any of the break-in phenomena creates sonic differences is the same as saying changing the thermostat by 1 degree is a change in sound. Technically true if you have a microphone to measure the tiny changes, not true if using ears to try to assess the tiny changes. I'm sure there is some corner case driver out there that actually has dramatic break-in, would be nice to see actual measurements since measurements are so easy to do. For the moderately large range of drivers I have experience with, no way anybody is going to hear the difference after first use.
Of course, the What Hifi interview drops the suggestion that the sound always gets
better after run-in, which is a
trick right out of the carnival meant to keep the audio mystery alive and prevent returns of bad sounding speakers from customers who bought the wrong gear.