Hitch mounted cargo carriers

No doubt gear in the carrier will be dustier than gear inside the car but I don't think that's the reason not to get it. You should just try and put things in the cargo carrier that are less sensitive to dirt and dust. Another possibility is to purchase a cargo box which is sealed and hence gives better protection.
Another solution in a different direction would be a roof top carrier but then you get into the problem of loading it. They aren't very convenient.

Anyway, good luck with whatever you decide on getting. 

Yes - it does get dirty I'm afraid but if you are careful with packing and (if need be) wrapping things up then you wont have too many problems.

Roof carriers are difficult to load and its not easy to get funny shaped items into them

There's no perfect solution but the rear carrier works well for most people