On ships, planes, and spacecraft, the laws of the country the vessel is registered in apply. That's why a whole lot of cargo and cruise ships are registered in Panama or Liberia, because lax labor and tax laws, and the country of the person injured gets second. So who gets to prosecute first depends on which side of the ISS the crime takes place on -- if it's on the Russian side, they get it, if an American did the crime or was crimed upon, they try the wrongdoer after the Russians do. And then there's weird-ass shit like when an American shot another American at a research station on an iceberg in a part of the Arctic ocean claimed by Canada, Canada noped out and gave it to Denmark, as Greenland was the killer's first port of call after the act, and since it wasn't a ship the US didn't want to take authority over it, so it went to the US, but which state to hold the trial in was a similar clusterfuck about jurisdiction ...
https://www.youtube.com/watch?v=T6E2VfyRR1Y