Movement of air beginning to bring chill? Not the first (6)
I believe the answer is:
breeze
'movement of air' is the definition.
(air current)
'beginning to bring chill? not the first' is the wordplay.
'beginning to' suggests taking the first letters.
'chill?' becomes 'freeze'.
'not the first' means to remove the first letter.
The initial letter of 'bring' is 'b'.
'freeze' with its first letter taken off is 'reeze'.
'b'+'reeze'='BREEZE'
(Other definitions for breeze that I've seen before include "Faint stirring of the air" , "Air current" , "Minor blow" , "Bit of a blow" , "The building block is light" .)