not all crimes qualify for extradition, it’s usually to do with the severity of the crime
some international bodies may be involved iirc, like interpol and Europol etc
it can also be done by three letter agencies or even just normal police. The extradition is usually just burocracy that slowly escalates up the levels of justice and finally to the indicting country
geopolitics can make this extremely complicated, because countries never cooperate out of goodwill, usually there’s underlying treaties that bind a bunch of them to commit to extradition
usually criminals get away with something, because the countries they live in don’t cooperate with an extradition in various ways
because it’s international and countries act in their own best interest, jurisdictional boundaries are sometimes extremely weird and assumed jurisdiction can make things difficult (multiple charges for a crime, countries not extraditing because they don’t trust the judicial branch of another country, weird quirks of things like law in space and maritime law, …
A few things to add to that: