driver moving too much

Find out if the problem is feedback or record warp/turntable defect by putting the pickup down on a record while the table is stopped. Tap the record and see what happens.
It certainly is a good idea to have the table on a solid base so that vibration does not get applied to it, although some tables have excellent vibration damping and are surprisingly immune to this. I have found that the worst accoustic feedback comes from the record itself acting as a diaphram excited by airborne sound. A record clamp helps this as does a good cover on the table.
The arm/pickup combo greatly affects the situation. Since I began using a Sony PSX800 table with linear "Biotracer" arm I have had absolutely no problem with accoustic feedback. The biotracer arm is servo-controlled, and has a completely massless "limp noodle" feel to it.

I had the same problem with my Linn LP-12 Grado combination. If it is not a feedback issue, it can be as in my case a compliance mismatch with the tonearm and cartridge. I put on a new Shure V15vxmr and its 95% better!I have an article written by John Grado in the 70's that explains tonearm compliance further. It is kind of out there. I can fax it next week if you would like to read it.

Looks alarming doesn't it? I've been coping with this phenomemon for decades and doubt it does any harm. I asked a trusted friend about it and he said all turntables exhibit woofer pumping to one degree or another. Perhaps Mara can address the electrical forces that make it move though it makes no sound.

I suppose you could construct a vacuum bubble or build a bunker for the table but let's keep it real. I had good luck raising my turntable about 18 inches above the tops of the speakers. It reduced the erratic woofer pumping quite a bit but did not eliminate it. Find a less sonically busy place to put your table. Or try the head in the sand approach and leave the grill covers on.

Twl gives good advice. I solved a problem with floor-borne vibrations when I had my rega 3 on a sideboard by getting a slab of 3 inch thick firm foam, placing a 2 inch thick concrete slab onto the foam (compressed the foam to about 1.5 inches) and then putting the turntable on the concrete. This completely cured the floor vibrations from feeding back to the turntable and got rid of the rumble.

I would say that floor vibrations are only a problem if your room has a suspended floor.

a warped record will let the speaker driver dance even on moderate and low volumes in my system.

The system is responding to subsonic frequencies. These can be generated by warps in records, turntable rumble, or mismatches in tonearm/cartridge combinations. Or it is possible that the system is not capable of reproducing certain low bass frequencies, and is unloading the bass-reflex cabinet, causing the woofer to bounce because the air mass in the cabinet is unloading. Or it could be acoustic vibrations coming up the stand,into your turntable Or it could be a combination of all of these things.

If you notice that it only happens on certain records, and not on others, it is probably a warped or defective record. If it happens more when you play loud, it is probably acoustic vibration entering the TT.

I'd try moving the turntable into a different spot in the room, where there may be less bass frequency buildup. If the TT is behind the speaker, especially in a corner, this is bad placement.