run down vs follow up on

run down and follow up on both are verbs.

Word	Noun	Adjective	Verb	Adverb
run down	No	No	Yes	No
follow up on	No	No	Yes	No

As verbs, follow up on is a hypernym of run down; that is, follow up on is a word with a broader meaning than run down:

Other hypernyms of run down include act on, pursue.

run down (verb)	follow up on (verb)
use up all one's strength and energy and stop working	carry further or advance
injure or kill by knocking (someone or something) down and passing over the body, as with a vehicle
pursue until captured
move downward
examine hastily
deplete
trace