The dogs ran off, being self-willed (10)
I believe the answer is:
headstrong
'self-willed' is the definition.
('headstrong' can be a synonym of 'self-willed')
'the dogs ran off' is the wordplay.
'off' indicates anagramming the letters.
'the'+'dogs'+'ran'='thedogsran'
'thedogsran' with letters rearranged gives 'HEADSTRONG'.
'being' is the link.
(Other definitions for headstrong that I've seen before include "obstinately self-willed" , "pig-headed" , "Obstinate" , "Wilfully determined" , "Showing wilfulness" .)