Black dog who gets around? (3-6)
I believe the answer is:
jet-setter
'who gets around?' is the definition.
'black dog' is the wordplay.
'black' becomes 'jet' ('jet' can be similar in meaning to 'black').
'dog' becomes 'setter' (setter is a kind of dog**).
'jet'+'setter'='JET-SETTER'
(Other definitions for jet-setter that I've seen before include "Member of the rich and famous who travel widely" , "Person who travels widely and frequently for pleasure" , "Rich and fashionable traveller" .)