Dahlquist DQ10 issues

I have a set of Dahlquist DQ-10 that sound great, but if I listen at moderate volume for 30 minutes or so the right one begins to fail. It almost sounds like a completely blown speaker.  when I turn the system off and go to use it again later, it seems fine. If I listen at lower volumes, it seems to work just fine for as long as I want.

I thought perhaps I had an amplifier issue, so I tested by listening until I started to hear the failure, then replacing with another speaker and it was fine (I will hook up to a different amplifier as well, but I’m fairly confident it’s the speaker).

is it possible there are components in the speaker that are overheating that are causing an issue?  i’ve never had symptoms like this with any other speakers.
jleluga, Swap channels with the speakers. I would like to make sure the same speaker fails hooked to the opposite channel. All you proved is that the good speaker is ...good. 
Assuming it is the speaker my guess would be that you have to rebuild the crossovers. You want both channels to be exactly the same so you will have to rebuild both of them by replacing components with like parts. It is the capacitors that need to be changed followed by resistors. Coils do not fail unless somebody cuts the wires.
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Your DQ-10 pair was manufactured with metalized Mylar capacitors.    Film caps don’t age, dry out or leak as do aluminum electrolytics.     Unless a resistor burns: they don’t typically go bad either.    If there were an issue with the crossover, or the drivers: it wouldn’t be intermittent.     That’s: unless there’s a loose connection or corrosion (ie: a wire loose at the 5-way, corrosion in the fuse holder, etc).    Next time the symptom arises, swap the speaker cables between channels (L & R, either at the amp or speakers) and when the problem follows: you’ll know if it’s upstream (a cable or component fault).
btw: If the system’s still putting out some sound, but at (seemingly) much lower levels: try cleaning the treble level control (and again, the tweeter fuse holder).      When the upper freqs are missing, everything sounds way off.      I suppose I could have mentioned the 80uF/50V electrolytic, that blocks highs from the mid-woofer, but- if that were bad: it wouldn’t likely be intermittent (the 4" Phillips just wouldn’t be working).
@rodman99999 , it sounds like a thermal issue. Don't you think that the most likely place for a thermal issue is the crossover? Certainly potentiometers and L pads are prone to causing problems. jleluga, you can clean the potentiometers with freon solvent which you can buy as electronics cleaner https://shop.advanceautoparts.com/p/crc-qd-electronic-cleaner-11-wt.-oz.-05103/7070074-p?product_cha...
You just point the tube at any open section and blast away.
I'm thinking an overheated voice coil. 
Could be a rubbing voice coil on that woofer. Did you ever have the foam surrounds replaced on both woofers? 
"...so I tested by listening until I started to hear the failure...".                                                      Knowing how that, "failure" sounds would be most advantageous, to anyone attempting a diagnosis.      ie: A dead, complete speaker system (bad connection), rattling lows (surrounds & rubbing VC), highs gone (possibilities mentioned above), etc.