Yes RV refrigerators do work while traveling down the road although they're less efficient and don't work as good when traveling down the road.
RV refrigerators can be either run or propane or electricity and sometimes battery power and I would recommend using the battery power for the RV refrigerator while traveling down the road.
But still you can use the propane to keep the RV refrigerator going if you prefer to do so.
RV refrigerators work best when the RV is level and the RV refrigerator is also level but there's not a problem with keeping your RV refrigerator going while going down the road.
But usually just running the refrigerator enough to cool down before you drive off is usually enough to keep the food and RV refrigerator cold enough until you get to where you're going and can plug the RV back in.