Quite simply, that's a usage of the -te form; connecting two verbs. All your examples have a nuance of "cause and effect".
did it influenced (by something/someone)
did it by being threatened (= was/were threatened into doing it)
did it by being asked to (= did it at one's request)
As a rare case, やる could be an auxiliary verb, meaning "to give a favor".
相手を勝たせるために、試合でわざと投げられてやった。
To let the opponent win, I was intentionally thrown by him in the competition (in a Judo match or something).