down the stairs vs on a lower floor

down the stairs and on a lower floor both are adverbs.

Word	Noun	Adjective	Verb	Adverb
down the stairs	No	No	No	Yes
on a lower floor	No	No	No	Yes

As adverbs, down the stairs and on a lower floor are synonyms defined as:

Other synonyms of down the stairs include below, downstairs.

down the stairs (adverb)	on a lower floor (adverb)
on a floor below	on a floor below