I have an on board mover which works fine for me on wet ground. As it drives the wheels directly it's no different to a car. I expect that there are limitations of course. If it's so wet that it's started to sink then it would probably struggle. The main problem will be the jockey wheel on soft ground as it will dig in quite easily. It the ground is firm but just wet and fairly level then I would think that you should be OK.
Perhaps others have had different experiences.
Just for the record I have a Powrtouch mover on a Sterling Europa 500.
Derek, I don't think you can use snow chains on caravan wheels with movers fitted, they rely on contact with the tyres. These type of movers would be more effective than using your car, or on very soft ground, use a type of matting to spread the load.