Delicate creature, get down dog! (11)
I believe the answer is:
swallowtail
'delicate creature' is the definition.
'down dog' is the wordplay.
'down' becomes 'swallow' (I've seen this before**).
'dog' becomes 'tail' ('tail' can be a synonym of 'dog').
'swallow'+'tail'='SWALLOWTAIL'
'get' is the link.
(Other definitions for swallowtail that I've seen before include "In circle, mythical snake does" , "Butterfly" , "coat" , "insect" .)