We try murderers from the US - Same as the EU- same as other countries where the justice system sucks.
Why -where - how does a US Citizen get the right to be tried in the US when the crime was committed in another country?
If the crime is not a violation of international law, but only a violation of the law of the land they are in, they retain, or at least, SHOULD retain, the rights they enjoy as defined by the government of their home country. In some cases, extradition, yes. In other cases, tried at the location of the crime, but with a lawyer provided by the US embassy. There is no right or wrong, blanket answer to this, it's all based on the details and variables involved. I mean, look, would you want to be a tourist simply ACCUSED of murder in, say, Mexico city, and from there, simply subject to their justice system, their police work, etc? Innocent until proven guilty is not an internationally accept idea. Likewise, someone accused of terrorism in France, should NOT get a get out of jail free card to skip back on home where they'll likely not only not be convicted, but will get a promotion, as a result.
Terrorism is inherently an act of war, while murder isn't. Commit an act of war, no MATTER WHAT soil you are on, and you should be shipped out to face a trial NOT by a jury of your peers, a MILITARY trial.
Commit murder, your situation will need to be reviewed on an individual basis, and a determination made based on the details of that situation.