We've been running propane while on the road since 2007. (First propane/110VAC fridge)
Just shut it off during refuelings and, so I've heard, passing through tunnels that require you to have it off.
My brother had a problem keeping the flame lit in his on the road - he added a small baffle between the firebox and the outside vent - it cured the issue.
A 12V fridge pulls 9-12 amps, so the line between your tow vehicle & fridge would need to be adequately sized, including compensating for distance and connector resistances.
I've always enjoyed the freedom of running propane on the road.