I've always wondered, how exactly does it work when someone is trying to evade police and crosses the border to another country/state?

Do the police have no jurisdiction there? So even if they are JUST about to catch up to someone who has done some horrible crime, they have to stop chasing, even if they haven't alerted the other country yet/they haven't arrived yet?

How does this work with state borders as well?
They'll usually alert that country's authorities that there's a dangerous criminal within their borders.
I dunno about the rest of the world, but Police who work on the border in Victoria, Australia are sworn in as special constables for the state that they border on to, and they then have some jurisdiction in that state
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When you cross city/county lines they alert the local authorities as it's happening and they determine how to handle it from there

when you cross a state line in pursuit (in the US), it becomes a felony, the local authorities are alerted, and eventually the federal government busts your ass.
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here is the thing TS if someone is planning on crossing into another country.

1.The police wont have jurisdiction in that country unless some prior agreement has been reach between those 2 countries. If you do get caught the arresting officers while always be sworn officers of the country you have decided to run to. They will then hand you over to
Your own countries police officers to escort you back.

2.Depending on what country you go to, an extradition treaty might been in place between those 2 countries which will make it easier to have you sent back to the country which you committed the crimes in.
3. as I see you're a Kiwi It would not be much of a hassle to extradite me back to Australia if I went to NZ after I committed crimes in Oz.
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There wouldn't really be any point in the word 'extradite' if people never got in trouble for crossing borders after committing crimes, either.
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Insightful answers, but what I REALLY mean is a situation like this-

Some horrible person has killed a bunch of people, and is making a getaway. The police are alerted immediately and are there straight away about to take him down, but he crosses over the border and gets away because that countries authorities aren't near enough yet.

Even if some horrible criminal is on the verge of getting caught, do they have to give up the second he leaves jurisdiction, even if it means they will get away?
Yes, they don't have jurisdiction. Operations on foreign soil is basically frowned upon.

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