I would expect that a moving pickup would be akin to using different woods on the instrument. If you were to attach a string at either end to an infinitely dense and immobile object, you'd get the perfect vibration of the string. To me, this was close to what I got when I played a neck-through Modulus many years ago. The impact of different woods on tone occurs when those woods vibrate in characteristic ways to absorb some frequencies of the string's vibration.
Since the pickup is detecting the relative movement of the string, any movement of the pickup will impact the output. If the pickup could be made to move with a particular frequency, then it could act as a filter centered on that frequency. If the phases were in sync, the pickup should attenuate that frequency. On the other hand, reverse the phase and that frequency should increase in magnitude. The reality is that you are unlikely to get a consistent vibration of your pickup and any movement will have an impact on all frequencies. The more significant the movement, the more significant the impact on tone.
With tone woods, their impact on the output will often be a positive one. I suppose a little positive impact is possible from serendipitous pickup movement. Personally, i'd rather have it locked down to the instrument. After all, whenever I am looking at a bass to consider buying it, the very first thing I do is play it unplugged to see what sort of natural vibrating tone it has. If I like that, then wouldn't that be what I want to amplify?
-bob
