Conservative is in rush to become whip (8)
I believe the answer is:
chastise
'whip' is the definition.
(thesaurus)
'conservative is in rush' is the wordplay.
'conservative' becomes 'C' (UK political abbreviation).
'in' indicates putting letters inside.
'rush' becomes 'haste' (synonyms).
'is' going into 'haste' is 'hastise'.
'c'+'hastise'='CHASTISE'
'to become' is the link.
(Other definitions for chastise that I've seen before include "Whip perhaps" , "Berate" , "Scold severely" , "Inflict physical corrective treatment" , "Punished severely" .)