move back vs back down

move back and back down both are verbs.

Word	Noun	Adjective	Verb	Adverb
move back	No	No	Yes	No
back down	No	No	Yes	No

As verbs, back down is a hyponym of move back; that is, back down is a word with a more specific, narrower meaning than move back:

Other hyponyms of move back include fall back, retreat, retrograde, back off, back up.

move back (verb)	back down (verb)
pull back or move away or backward	move backwards from a certain position
	remove oneself from an obligation