Something that sort of came to light a short while back when Amir gave the Elac UniFi 2.0 the run through is that with digital formats (and possibly depending on the music server program) the files (whether locally stored or web-streamed) can contain strong DC or near-DC (infrasonic) components, and that if you play such a file using an amplifier that passes DC ("dc-coupled") that the signal you send to the speaker will have a DC component, possibly a strong DC component.
Asking other people if they have the same problem with the same file is only likely to cause confusion, as has occurred in this thread. The other person playing the same file may be using an amplifier that blocks infrasonic frequencies. Or this may be done in the music server software they are using, or at some layer in the audio infrastructure on the computer, or possibly the DAC, etc. Even if another person says they see what seems to be the same as what you see, what do you then conclude? And if they say that they don't see it, what do you conclude?
What needs to happen is first some thought needs to be given to the unexplained motion of the speaker cone, specifically with respect to whether is seems to correspond to infrasonic content or even DC. If the hypothesis is that this is what is going on, then the file needs to be examined using some kind of spectrum analyzer, to see whether it does in fact have infrasonic or DC content. If it does, then you have your answer but still need to decide what you should do about it. If the spectrum analyzer does not confirm the presence of infrasonic or DC content but this is what your eyes tell you as you watch the cone, then you've got a mystery on your hands.