race vs push forward

race is a noun but push forward is not a noun.

Word	Noun	Adjective	Verb	Adverb
race	Yes	No	Yes	No
push forward	No	No	Yes	No

As verbs, push forward is a hyponym of race; that is, push forward is a word with a more specific, narrower meaning than race:

Other hyponyms of race include barge, thrust ahead, buck, charge, shoot, shoot down, tear, dart, dash, flash, scoot, scud, shoot.

race (noun)	push forward (noun)
a canal for a current of water
a contest of speed
any competition
people who are believed to belong to the same genetic stock
(biology) a taxonomic group that is a division of a species; usually arises as a consequence of geographical isolation within a species
the flow of air that is driven backwards by an aircraft propeller

race (verb)	push forward (verb)
compete in a race	push one's way
move hurridly
cause to move fast or to rush or race
to work as fast as possible towards a goal, sometimes in competition with others