pull back vs fall back

pull back and fall back both are verbs.

Word	Noun	Adjective	Verb	Adverb
pull back	No	No	Yes	No
fall back	No	No	Yes	No

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

Other hyponyms of pull back include retreat, retrograde, back down, back off, back up.

pull back (verb)	fall back (verb)
make a retreat from an earlier commitment or activity	go back to bad behavior
move to a rearward position; pull towards the back	retreat
stretch back a bowstring (on an archer's bow)	hang (back) or fall (behind) in movement, progress, development, etc.
use a surgical instrument to hold open (the edges of a wound or an organ)	fall backwards and down
pull back or move away or backward	move back and away from
	have recourse to