You mean what would happen if for example an Austrian archduke where to be assassinated by a Serb activist whilst visiting Bosnia?

Read up on Franz Ferdinand if you want to know how it goes.

"American Assassinated Putin" or "Russian Dissident Assassinated Putin"

I think both of these headlines would be printed. Russia would blame US, and US would blame them.

Four, "High Ranking" Democrat senators might agree to send an email.

"Asking for a friend"

Thoughts and prayers

It would be a gift to whatever country was hosting the shooter. They could disown them completely. Russia in this case would blame the hosting country for not being good enough security wise and there would be a flaming hot discussion. Possibly some level of conflict and then it would die down.

Why would they extradite that person? The murder took place in country A and the leader of country B got killed. That's a problem for the courts in country A.

Some countries have a legal proviso that can activate their court system for crimes that didn't happen within their jurisdiction. Some Syrian torturers that ended up fleeing Syria too found themselves in courtrooms in the EU. So if the perpetrator was able to kill leader of B in A but then fled the country, if B somehow got ahold of them, they could prosecute their leader's murder on their own turf. That's assuming they wouldn't just send a hit squad.

All the headlines would just say things along the lines of "B-born naturalized citizen of A killed B leader." I think it's similar to all the al qaeda or IS terrorists who often held passports of the countries they terrorized.

The relations between these countries would obviously be strained. But how strained would depend on more factors. Where did security break down, who fucked up, and how did everyone react to this assassination? If B was indeed Russia and the shooter was Russian-born, you can bet they will come up with a media narrative how the culprit was long sought for gulag internment for their wild ideas but managed to flee from the glorious motherland. And then duped the hapless people of A to give them citizenship, the stupid, naive fools! If only they had listened to us, who are always right, before! Then this tragedy could have been averted.

Putin is a war criminal with an arrest warrant by the ICC. He ain't visiting nowhere, apart from shithole countries that aren't part of ICC.

Case by case basis.

So much would depend on the countries involved, and their current relationships.

Let's say it's a former Russian citizen who kills Putin in the USA. The new Russian leader might want to have the killer tried in Russia, or they might want to throw the whole thing to the Americans. Legally, if the murder takes place in the USA, it's the duty of the individual state to deal with it. But if the US President declares it an act of international terrorism, they could push through sending him home.

That's not even getting into how the murder was done. Putin would have his own security team, plus Secret Service protection, so how did the killer get past them? Was there a conspiracy?

There's no single answer possible.

All of the above. Every single group will twist it to suit their needs, but those who control the narrative will ultimately win out