Woofer pumping - what gives?

The above answer is correct. The reason you don't see this with the tapes is that they don't go below 20HZ. You are seeing arm/cartridge resonance because the effective mass of your arm is too high for the compliance of your cartridge. This sub-sonic "signal" is actually using up a lot of your amplifier's power reserve. You could reduce this by buying a lower compliance cartridge or, if it doesn't bother you and you don't notice it, just live with it.

Skn77a, I have the same situation but to a lesser degree. Please explain what you mean by the effective mass of the arm is too high for the compliance of the cartridge. I would like to eliminate this woofer excursion.
Thanks,

Ditto Sean. This is why some (esp. older) integrateds/pre's have a "low freq filter" or something similar (they all are hi-pass filters attenuating the signal from ~ 20 Hz and below). If you're sure your cartridge is OK, maybe your rig has this feature - use it. If not, you may want to add such a filter, although its proper implementation is yet another entire story.

Could it also be low frequency feedback from the speakers to the turntable ? If your turntable is not well isolated from the speakers you'll get this problem (I did a few years back). If your speakers are spiked into a wooden floor (either directly or through a stand) and if your turntable is on a rack or piece of furniture that is also sat on the same wooden floor then I bet you've got feedback.
I solved it two ways : either wall mount the turntable, or place the turntable on an isolation platform. I made my own isolation platform from a concrete slab and a thick wedge of foam. In either case the "wobbling woofers" were completely cured.
The above answers are also good reasons for your situation, but if your setup fits my description then try a simple isolation platform to see if it makes a difference.

If your turntable has a dustcover, I'd suggest removing it–that could exacerbate low-frequency "feedback" from your speakers.