This is a continuance of this question regarding Interpol.
The answer to that question left a hole; who's going to be the international pursuer in the classic, world-wide cat-and-mouse game? It is a trope in media that there's a single agent in charge of catching some criminal, who'll pursue them to all corners of the earth. This trope is popular because it gives the pursuit a far more personal edge. I always thought such agents would be from, or authorized by, Interpol. Apparently however, Interpol does not have agents.
However, I'd still think there is some room for officers from other countries to take part in pursuits in different countries, especially when the countries involved are perhaps a part of some international alliance of sorts (like Interpol).
Let's say there is an agent who is an expert on some international criminal, who comes from country A. The criminal is presumably in country B, and both country A and B are cooperative in the relevant ways. Would it be possible and plausible for law enforcement/the government in country B to authorize the agent to pursue the criminal in country B? If so, how would this process take place? I reckon there is a lot of variability and details, so any referrals to places where I can read up on various specific cases would be nice.