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

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!!