The Republicans are up only at first, to move up and down but go nowhere (4)
I believe the answer is:
pogo
'move up and down but go nowhere' is the definition.
I can't judge whether this defines the answer.
'the republicans are up only at first' is the wordplay.
'the republicans' becomes 'gop' (the Grand Old Party, nickname for US Republicans).
'are up' says the letters should be written backwards (in down clue: letters go upwards).
'at first' suggests taking the first letters.
The initial letter of 'only' is 'o'.
'gop' written backwards gives 'pog'.
'pog'+'o'='POGO'
'to' is the link.
Can you help me to learn more?
(Other definitions for pogo that I've seen before include "Punk movement" , "-- stick" , "gets jumped on by children" , "punk dancing" , "jump up and down" .)