For more generality, let us assume Bob committed $SOMECRIME in country A and fled to country B. For simplicity, assume that Bob was apprehended in country A and there is no risk he will escape A’s custody before extradition proceedings are over (either through legal means if they are not charged in time, or non-legal means if they break free).
Extradition: basic framework
Country A request that country B hands over Bob. (That makes A the requesting country and B the requested country, in extradition parlance.)
In theory, B could decide whether to extrade Bob in any way they please in accordance to their internal legal order. In practice, many things are similar across countries, and often codified into treaties. I will use the example of france, its internal provisions for extradition from CPP 696-4, and its extradition treaty with the United States; but unless mentioned otherwise, I believe the general principles are valid in most Western jurisdictions.
- B almost always requires that the extradition requests lists the charges that country A intends to prosecute Bob for. Recently, charges of campaign finance violations against Sam Bankman-Fried were dropped, because they were not included in the extradition request that the USA sent to the Bahamas and that the Bahamas accepted. (The US government could probably charge SBF for those charges and courts would not reject them, but then nobody would extradite anyone to the US ever again.)
- B almost always requires that $SOMECRIME be also a crime in B’s jurisdiction, usually with a minimum sentence (one year of prison in the US-France treaty, article 2§1). Murder is a crime pretty much everywhere, but what about blasphemy? Holocaust denial? Selling food to Cuba? Selling weapons to Iran?
- If B is a vaguely democratic country, Bob can fight extradition in B’s courts, and the judge can refuse extradition. Most (all?) democratic countries refuse extradition for political crimes: article 4 in the US-France treaty, and CPP 696-4 §2 in French law ("extradition is not granted... when circumstances show that extradition is requested on political grounds"). When it comes to extradition to China, Russia and other similar countries, France also has CPP 696-4§7 ("extradition is not granted... when the requested person would be judged (...) by a court that does not guarantee (...) the fundamental rights of defense")
- B can sometimes request that A promise to not apply certain punishments. The typical case is death penalty (article 7 of the US-France treaty), but some jurisdictions apply corporal punishment (e.g. whipping). In france, extraditing a person facing such penalties would violate CPP 696-4 §6 ("extradition is not granted... when the crime that justifies the request is punished by the requesting state by a penalty contrary to French public order").
- A few countries (including france, CPP 696-4 §1) refuse to extradite their own nationals. If you read article 3 of the US-France treaty, you might be under the impression that both countries reserve the right to refuse extradition on that basis. However, I would guess the practice is very unbalanced - if the US doesn’t care it will happily extradite its own citizens; but it knows France does care, and will not even bother requesting extradition of a French citizen.
If Bob is extradited, he will be forcibly moved back to country A, where courts can pick up the case in the usual fashion.
What if that fails?
Evading A’s jurisdiction does not always mean that Bob will go free. In the US-France case, if extradition is denied purely based on nationality, the requested country must prosecute Bob (article 3§2). (France claims a rather extensive jurisdiction, whereby anything done by a French citizen everywhere is subject to French penal law under CP 113-6. They even claim jurisdiction when the victim is a French citizen, CP 113-7! Subject to some limitations depending on gravity etc., but still.)
Prosecution in country B, willing or not, might still pose numerous practical problems, as most of the evidence of the crime is from country A:
- country A might refuse to cooperate with country B's law enforcement;
- forensics, witness testimony etc. might involve logistical difficulties;
- law enforcement standards might be wildly different in country A and country B. For instance, US courts require oral evidence for testimonies; a US criminal case that goes to trial will (almost?) always testimony from a police officer, and cross-examination by the defense. French courts do not work that way at all; the vast majority of French police officers never appear at trial, and their "testimony" comes purely in the form of a police report. A French police officer testifying in US courts would probably make a fool of themselves during cross-examination; a US police report would probably be seen as too short/unclear in French courts.