We had a 1996 Mazda...it was called a Ford Contour. 5 spd stick and a great toad. Around 130,000 it started doing the nickel and dime dance. The 130,000 did NOT include the towing mileage as it did not register on the odometer. We went to a nearby Mazda dealer in 2007.
The dealer tried to help us, but could not. ANY Mazda could not be towed 4 down. The dealer even called some engineering dept...and they also said no Mazda could be towed 4 down.
That included manual transmission. So...we went to the Chevy dealer and bought a 2007 HHR automatic. Towed 4 down with no problems. Pull one fuse and go. Really nice car.
Our current toad is a 2011 Ford Ranger 4WD automatic. Had to buy a FORD ONLY "Transfer Case Disconnect Kit" ($400 installed)..but the pickup tows just fine.
I guess if you must have a Mazda, check out the dollies or car trailers. But most important, any car, read the owners' manual before you buy.