Leader of some parties, even Democrats, went quickly (4)
I believe the answer is:
sped
'went quickly' is the definition.
(I've seen this before)
'leader of some parties even democrats' is the wordplay.
'leader of' suggests taking the first letters.
The first letters of 'some parties even democrats' is 'SPED'.
(Other definitions for sped that I've seen before include "was 8 [FLEET]" , "Hastened" , "Went fast" , "Moved fast" , "Went quickly" .)