Is there any reason to think that the wood & cloth structure would have problems in the hot, arid weather of the Southwest?

(It would remain hangared--not left outside.)

Are you thinking of a permanent move, or just for a few days or weeks?

The latter would probably cause no problems, but I suspect a whole season or more in drier conditions will cause a little wood shrinkage which would require some careful tightening of various fasteners.
Problem areas -- propeller bolts, main spar bolts, horizontal & vertical stabilizer bolts, outrigger socket bolts.

There may be more.
My Australian Fournier lived first in northern France, and then in Victoria, both damp, rainy places.
Then it came over to our semi-arid Western Australia.
Obviously, I loosen and re-torque the prop bolts several times throughout the year (who doesn't?) but those other bolts & Allen screws have all needed a tiny bit of tightening up over the years.
We're talking turning them just one flat or so.

You can tell when careful tighteing is needed by rocking the top of vertical stabilizer back and forth, and waggling the horizontal stabilizer up and down and back & forth (both quite vigorously). See what they're like now, and then watch for changes.
For the main spar bolts, on an RF4 you can look down vertically in the cockpit and see when the main fuselage bulkhead pulls away a bit from the spar -- a tiny gap opens up. If you leave it, you then notice a gradually increasing 'phantom clonk' during rolling manoeuvres as the bolts wear in the fuselage holes, elongating them.
There is s 'simple' repair scheme -- see elsewhere on this site, but it does involve splitting wing & fuselage, so it's far better to keep those bolts snugged up than to wait for the 'klonk'.

I hope this is of some help.

Yours, Bob

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My problem is the opposite, permanent move from a location where my RF5B lived for the last 38 years with yearly average 60% humidity, semi arrid inland location at 3000' altitude to 72% humidity Istanbul by a lake at sea level. What to do?

Regards

Kadir