What circumstances are those? There are a ton of homeless Americans where I live and the embassy doesn't do anything until they commit a crime as far as I know.
If there is an act of war or natural disaster, the U.S. is good at extracting its citizens. There is also a decent precedent of negotiating to release people taken hostage or held prisoner by unfriendly regimes. (Or causing a fuss when Americans are harmed in a friendly country.) That, in turn, has a deterrent effect.
For Americans with access to legal counsel and the State Department, the benefits expand. Rich, overseas Americans thus present a unique free-rider problem.
Such a gruesome story: Kenyan sources supported Israel, and in the aftermath of the operation, Idi Amin issued orders to retaliate and slaughter several hundred Kenyans then present in Uganda. There were 245 Kenyans in Uganda killed and 3,000 fled.
https://en.wikipedia.org/wiki/Idi_Amin
Amin first escaped to Libya, where he stayed until 1980, and ultimately settled in Saudi Arabia, where the Saudi royal family allowed him sanctuary and paid him a generous subsidy in return for staying out of politics.[18] Amin lived for a number of years on the top two floors of the Novotel Hotel on Palestine Road in Jeddah.
https://www.imdb.com/title/tt0455590/
Movie The Last King of Scotland is Based on the events of the brutal Ugandan dictator Idi Amin's regime as seen by his personal physician during the 1970s.