I meant, does anyone investigate them end-to-end, and not just in their own country? I find it hard to believe that the FBI interacts well with law enforcement in, say, Poland.
well the RCMP and the FBI work together very closely, for one. An American FBI agent can get more done in Canada through RCMP connections than a rich and powerful Canadian can.
Yes, and vice versa, lets say the RCMP needs someone sent to Syria for 'questioning', then they'd phone the FBI and tip them off that such a person is coming to the US. And the FBI would promptly send them to Syria for 'questioning' of course also ignoring their citizenship and duty to deport people back to their country of citizenship.
I know it's hard to believe, but law enforcement of different countries indeed does cooperate really well, often even if these countries don't have very good diplomatic relationships.
Countries outside America do have law enforcement agencies for the most part, so yes.