fall back vs pull back

fall back and pull back both are verbs.

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

As verbs, pull back is a hypernym of fall back; that is, pull back is a word with a broader meaning than fall back:

Other hypernyms of fall back include draw back, move back, pull away, recede, retire, retreat, withdraw.

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