Lots and lots and lots of posts in the past on this subject in every forum category on RV.net. And I don't think there has ever been a general conclusion. There are those who do, and there are those who do not. Each person stating their opinion is very adamant about their own personal choice (and I am too).
There has been debate "out the wazzoo" about the safety factor, fueling at gas stations, and how long the refer still stay cold without being turned on.
The bottom line is, (as far as I've can tell), no one has "personally" had any mishaps or problems leaving the propane on or the refrigerator running when traveling. There have been lots of folks saying that someone else said ....
Documentation? Positive proof? There just isn't any, except maybe a single freak accident in the past.
Experts? Well, if you search the internet, you'll suggestions from all over and from all kinds of sources saying it's not a good idea, but still are not able to provide clear cut examples or some kind of documentation stating such.
In the end ... is YOUR choice.
Mechanically? Yes! you can run your propane and refrigerator when traveling. They are designed to do so.
Jesus never called anyone a "sinner".... did you know that? He did refer to the decisions people make as being "wise" or "not wise". I like that a lot better too! So the answer to your question rests in your own "wisdom." The choice is yours.