Synonyms for Move up:

n.

break into, join, land, start, fill in, arrive, attach, substitute, stand in, enter. bestride, burrow, cower, crane, crouch, balance, bend, crook, double over, bow. go forward (noun)
get ahead, prosper, advance, do well, go ahead.
move up (noun)
come up, uprise, go up, arise, rise, lift.
promotion (noun)
lift.

v.

advance (verb)
get ahead, go ahead.
ascend (verb)
arise, rise.
get up (verb)
uprise.
lift (verb)
come up.
motion (verb)
rise.
promote (verb)
advance.
rise (verb)
go up.

Other synonyms:

uprise
uprise.
Other relevant words:
advance.