What is the word that describes people who are easily influenced by propaganda or always do what all other people are seen to be doing without much thinking?
I know of no word quite so specific, but gullible describes anyone who is easily fooled.
Conformist / conforming - has the strong connotation of someone who "goes along" but not of that being a bad thing. One might (rightly) conform to a law one doesn't happen to like, or (wrongly) conform to a mob's violence or social injustice.
Suggestible - someone who takes on thoughts and ideas from others without conscious consideration
Susceptible (to propaganda) - can be influenced/affected
Influenceable (by propaganda) - as above
Malleable - bendable, implies resistance on some level that is ultimately suspended. The analogy is to metalworking.
Compliant - someone who complies over-readily
Sheepish - used in a certain sense, it means someone who follows a herd mentality. It can also mean shy or cowardly.
Lemming - (noun) means someone who follows a crowd even suicidally
I'd like to throw my entry into the ring. 'Weak Minded'. Wikipedia says for Weak Minded "... is the state of being easily impressionable or possessing a weak sense of self-will, judgement or conviction.
A weak minded individual's opinion may be easily swayed by propaganda or emotional manipulation tactics, as they do not possess an adequate ability to judge or discern the quality of an assertion, or they may exhibit a lack of discipline."