Move quickly as type of jazz brought up (4)
I believe the answer is:
dart
'move quickly' is the definition.
(I've seen this before)
'type of jazz brought up' is the wordplay.
'type of jazz' becomes 'trad' (I've seen this before).
'brought up' says the letters should be written backwards (down clues only).
'trad' backwards is 'DART'.
'as' is the link.
(Other definitions for dart that I've seen before include "Sudden quick movement" , "quarrel" , "Small narrow pointed missile" , "Tuck; arrow" , "Arrow - go fast" .)