Sorry, just old fashioned. The main issue with conventional auto boxes was the torque converter and the extra heat generated due to it changing up and down more often with antrailer on the back. Most manufactuers fitted coolers and provided you kept within the guidelines and checked the fluid regularly, there was never much of a problem. The CVT system uses an infinitely stepped system and at least in the old days, worled on belts, which could break. I am sure the modern systems are more technological, but I have no direct knowledge of them. I would suspect that provided you keep within the manufacturer's limits, then there should be no problem towing. As far as performance I suspect these will be similar to normal auto boxes in as such the fuel consumption will rise significantly due to the extra load.
Personally I like the torque converter system as it allows a 'creep' which is great in traffic jams. My own auto box also allows for the system to move to neutral if you are stationary for any length of time so stopping the generation of heat and wear, and reverts to 'drive' as soon as you release the brake.