Clue: Aviatrix, very briefly flying, ordered for this transport? (3,4)
Answer: AIR TAXI
Explanation: Aviatrix, very briefly flying, ordered for this transport? (3,4)
*(aiatrix), *with* "ordered" *as the anagram indicator*. 
"very briefly flying" *tells us to remove the *"v" *from *"aviatrix", *as it has flown* *(away).