Our Japanese red bean is air-flown from Hokkaido , Japan, and vacuum-packed to keep its authentic flavour.
This example was extracted from a brochure of MOF restaurant promoting various food items ranging from drinks to desserts. I don’t know why this strikes me as inappropriate. Perhaps it’s because I have never, from my experience , heard anyone saying “ this product is air-flown from Hokkaido”. Usually, I would hear people saying this product is imported from country X. Hence, to answer my doubt, I deceided to do consult various dictionaries to search for the definition of air-flown.
I did a quick check with my electronic dictionary –Oxford advanced Learner’s Dictionary.
Air-flown did not appear despite having typed the key word “air” or “fly”. But under the word “fly” , I got this reading:
6. (VN) to transport goods or passengers in an aircraft.

The MW dictionary says: intransitive verb
1 a : to move in or pass through the air with wings b : to move through the air or before the wind or through outer space c : to float, wave, or soar in the air
2 a : to take flight : flee b : to fade and disappear : vanish
3 a : to move, pass, or spread quickly b : to be moved with sudden extreme emotion c : to seem to pass quickly
4 : to become expended or dissipated rapidly
5 : to operate or travel in an airplane or spacecraft
6 : to work successfully : win popular acceptance
transitive verb
1 a : to cause to fly, float, or hang in the air b : to operate (as a balloon, aircraft, rocket, or spacecraft) in flight c : to journey over or through by flying
2 a : to flee or escape from b : avoid, shun
3 : to transport by aircraft or spacecraft

— fly at : to assail suddenly and violently

— fly blind : to fly an airplane solely by instruments

— fly high : to be elated

— fly in the face of or fly in the teeth of : to stand or act forthrightly or brazenly in defiance or contradiction of

The Encarta dictionary says:

fly [ flī ]


verb (past flew [ floo ], past participle flown [ flōn ], present participle fly·ing, 3rd person present singular flies)

Definition:

1. intransitive verb move through air: to travel through the air using wings or an engine


2. intransitive verb travel in aircraft: to travel in an aircraft


3. transitive verb take somebody or something by air: to take or send things or passengers in an aircraft


So,based on the definition given by the dictionary, the phrase "beans air-flown from hokkaido" is semantically fine but why does it sound so strange to use air-flown and why does it sound so much natural to say import? I dunno. Maybe I am just being weird.
I went to google the word air-flown and apparently, there were instances where we have examples like :
Fresh food flown to Britain
Air-freighted green beans
air-freighted organic products
broad beans flown in from abroad
Perhaps there is nothing worng with air-flown at all = ) .