Extortion and Blackmail
are well known tactics in Real Life
and in fiction to force unwilling compliance, with many variations. From an outside perspective, the risks are usually fairly even, or the request is outweighed by the threat; a life for a life, or money for career preservation.
However, sometimes the request far outweighs the threat, throwing the balance out of whack and resulting in this trope.
The blackmail in this case concerns something far more important than what the blackmailer is threatening; help us assassinate the king or we kill your wife. It's most often justified by threatening loved ones, leading the blackmailee to ignore the imbalance and help with the plan. It gets to headscratching levels when what is wanted will still likely or definitely result in the loss of what is threatened- it doesn't matter if you get your wife back when you just helped initiate World War Three.
Always Save the Girl
is a subtrope for love interests.
- Attempted in Olympus Has Fallen, where the villain tried to use the Presidents son to blackmail him into handing over a code that would detonate every American nuke, but was unable to grab him.
- In White House Down, the bad guys do succeed in grabbing the daughter of a police officer, which results in the President surrendering and the near nuclear obliteration of the Middle East.
- In Iron Man 3, the Vice President is willing to be a puppet leader in exchange for a cure for his daughters disability.
- On Series/Nikita, Nikita is threatened with the death of Michael unless she assassinates the President.