The volume of my speakers going down by itself and going back again in a minute or two

Without any further information I'd say it was demon possession, it could just a curse. Two Hail Marys and 5 Our Fathers for the curse, and get some Holy Water for the possession.. That should cover everything..

Then look at the sticking remote, and change batteries and clean the contacts if need be. Just a thought..

I'll put in a good word before sundown.. It is a day of rest.. :-)

Your new to you 20 year old pre and power amp might need a tech check.

I had the same thing happen. Then I realized the dog was stepping on the remote. All 7 pounds of him. 

Overheating? Turn off lights and look at tubes for red plating.

I'm thinking the possession thing. 
 

Both channels are experiencing the volume change? While I have never heard of such a thing... if it is in one channel, I would suspect a tube. ARC is really well made... so the most likely culperate is a tube.
 

ARC will have one tube for both channels or two or more per channel. If the problem is in one channel. Then switch the 2 or four tubes per channel in the Amp first and see if the channel changes... then try in the Preamp. If it is both channels, get a replacement and se if it goes away. 
 

You write: "I just upgraded a used Audio Research tube Preamp and a power amp." Do you mean that you upgraded to a used Audio Research preamp and amp ... or that you had the used preamp and amp upgraded?

Also, as obvious as this is, if the used equipment is new to you, do you have your old equipment? That would help isolate the problem. 

If you haven't already, you might try playing Mike Oldfield's Tubular Bells. What do you have to lose? 

noromance has a point, turn off the lights a see if ALL the valves get dimmer then brighter.

Do you need to bias the valves individually, or is it auto bias?

Are the small valves doing stupid things?

Did you pull the valves and then plug them back in? Stranger thing have happened.. None of this stuff cost anything.

Getting the holy water might cost you a fiver. :-)

Regards

Sounds like the new gear to me. 
 

  Send in and get caps replaced, looked at, cleaned, ther stuff replaced.

send or drive to factory if they allow it.

shipping sucks!

 Pay the money, that is some fine gear!!

Swap out amp or preamp to determine which piece is giving you the problem. If you are not a tech, then you're going to have to send that unit in for service. Could be something simple like oldhvy mentioned, worth checking.

is a built-in test sequence occuring?

If you weren't already an old hand at this, I would have suggested going back to first principles and cleaning/checking connectors and volume pots, but I'm sure you are way beyond that.  Good luck with the gremlins!

use a stopwatch, time it, always the same time delay?

My first inclination is there is an issue with the new to you preamp. If you have a spare pre try it and see if the problem goes away. If it does that at least narrows down the component. If indeed the preamp, and not knowing which specific audio research preamp you own, there could be a motorized volume control that is linked to a microprocessor. My suspicion would be the microprocessor control reaching temperature and doing some odd things making the motorized volume control turn fully down then back to normal. If it is a motorized volume control see if the volume knob is actually turning when the volume goes up and down. Some AR don't have a motorized volume control and are microprocessor controlled. I think the problem lies within the physical means for changing the volume within the unit like the motor, what controls the motor, or a microprocessor is out of spec. Only a guess though and if true that would probably require a journey to a tech to iron out. Best of luck.

Sure sounds like the electronics need checked out to me as well. Good luck with them and hope repairs are reasonable. Probably worth keeping!

Wow! Thanks for all the suggestions! I ought my Audio Research a few months ago from eBay and never had any issue with it. Just got the BAT 6200 from Audiogon yesterday and hooked everything last night. I did find one problem, though. One of the speaker fuse is broken. I am not sure if this cause all the trouble. I have a pair of Carver ALIII for over 20 years. It has two fuses each speaker. One for the base, one of ribbon. I did use one of my other pre-amp with BAT amp, I still found the sound went up an down some how. However, after I only hook up the speaker without the broken fuse, the weavy sound is gone. Very strange. Anyway, I will go to Homedepot to ge the fuse and and see this if this problem goes away after replace the new fuse. Ting

The  model of the Audio Research I bought is LS17SE Tube Hybrid Stereo Preamplifier.