The sole function of a damper (to give it a more accurate description) is to stop any bouncing after the vehicle has hit a bump or the spring has been compressed after going around a corner, so fitting dampers to the caravan and stronger dampers to the car will reduce the tendancy for snaking and bouncing.
Years ago I changed my boat trailer that had indespension units and would snake a bit at 60 mph for a converted caravan chassis that had coil springs and dampers, rock steady at 85mph!
This is one of the reasons that some 4x4's are less stable than they should be, they have quite weak damping to avoid "pump down" which is where the damper does not allow the suspension to return to its original position before hitting the next bump so the suspension remains compresed and hard. In my days of motocross racing I would adjust the damping on the bike according to how rough the track was. Recently I helped load a dingy onto the roof of a Discovery and was surprised how easy it was to rock the vehicle from side to side, 3 men could have probably got it rocking enough to tip it over. If I ever end up towing with a 4x4 the first thing I would do is bin the standard dampers.
On high mileage cars, even if the dampers pass the mot it is worth replacing them especially if you are towing, some organisations recommend every 40,000 miles. By the time we sold our VW Sharan with 115,000 miles on it's original dampers it passed it's MOT, but it was definitely not as stable as it was when it was new, both solo and towing.