[jccooper]

In 1985 the Supreme Court ruled that police are allowed to use deadly force to stop a fleeing suspect only if they have "probable cause to believe that the suspect poses a significant threat of death or serious bodily harm to the officer or others." It was soon tightened up to a standard of "objective reasonableness", so one can't simply "believe" in a threat.

Before 1985, using force, including lethal force, to stop a "fleeing felon" was accepted from common law--for any party, not just police. Lethal force would not be acceptable for minor or non-violent crimes. Doubtless the trope comes from practices pre-1985. I'd imagine the same provision of common law applied in the UK at one point, but I couldn't say when or how it stopped. Perhaps quite a long time ago.

Police and civilians also have legal justification for use of lethal force in defence of one's self or others from imminent use of deadly force (and usually other serious violent crimes like kidnapping or rape). Exactly how that line is drawn varies slightly; some states have a strong "duty to retreat" whenever possible, but most don't.