Leave domesticated animals to fly west with swallow (4,4)
I believe the answer is:
step down
'leave domesticated animals' is the definition.
'step down' can be an answer for 'leave' (stepping down is a kind of leaving). I'm not certain of the 'domesticated animals' bit.
'fly west with swallow' is the wordplay.
'fly west with' becomes 'step' (I can't justify this - if you can you should believe this answer much more).
'swallow' becomes 'down' (I've seen this before).
'step'+'down'='STEP-DOWN'
'to' is the link.
Can you help me to learn more?
(Other definitions for step down that I've seen before include "Give up office" , "Leave office" , "Resign from office" , "Give up or retire from a position" , "Get off one's soap-box" .)