Race involving everyone amongst Republicans (6)
I believe the answer is:
gallop
'race' is the definition.
(thesaurus)
'everyone amongst republicans' is the wordplay.
'everyone' becomes 'all' (all people).
'amongst' means one lot of letters goes inside another (I've seen this in other clues).
'republicans' becomes 'gop' (the Grand Old Party, nickname for US Republicans).
'all' going inside 'gop' is 'GALLOP'.
'involving' acts as a link.
(Other definitions for gallop that I've seen before include "Fastest gait of a horse" , "Log pal in at this pace" , "Fastest run of a horse" , "Faster than a canter" , "Move fast" .)