Everyone featuring in the Republican party race (6)
I believe the answer is:
gallop
'race' is the definition.
(thesaurus)
'everyone featuring in the republican' is the wordplay.
'everyone' becomes 'all' (all people).
'featuring in' means one lot of letters goes inside another.
'the republican' becomes 'gop' (the Grand Old Party, nickname for US Republicans).
'all' placed into 'gop' is 'GALLOP'.
'party' is the link.
(Other definitions for gallop that I've seen before include "All at GPO can go at this rate" , "Run fast like horse" , "Fast ride" , "'Run at a high speed, like a horse (6)'" , "Log pal in at this pace" .)