I know that many nations that don’t have the death penalty will refuse to extradite a criminal to the US if he/she faces a possible death sentence. The work around that’s usually arranged in such cases is that the prosecuting jurisdiction agrees not to seek the death penalty, and the subject is then extradited.
Let’s imagine a scenario where a suspect goes on a multi-state killing spree and then flees to France. If one of the states (let’s say New York) agreed to waive the death penalty, but the other states (let’s say Ohio and Virginia) did not, could the suspect be extradited to New York? Assuming he was extradited from France, could Ohio and Virginia then in turn seek extradition from New York, with the death penalty still on the table in their states?
IIRC extradition agreements limit the effect of the extradition to only those charges laid down in the extradition request.
Of course, if this was ignored there would be little the extraditing state could do about that matter other than complain, but it would make unlikely the success of any future extradition request for other people.
Most extradition agreements include a clause of specialty that basically says what Pjen mentioned: The accused may only be prosecuted for the crimes laid down in the request and for which the extraditing state has granted extradition. For an example of this doctrine in U.S. law, see the case of Rauscher, 119 US 407 (1886) about a seamn extradited from the US to Great Britain. The Supreme Court held that it was “unreasonable that the country of the asylum should be expected to deliver up such person to be dealt with by the demanding government without any limitation.”
In your case, France would probably check which crimes the refugee is accused of in the U.S. If there are several jurisdictions involved, it would seek to get a guarantee from any one of them that the refugee would not face the death penalty. If one state refused to do this, France would very likely deny extradition and try the refugee according to its own laws.
Also keep in mind that from the foreign perspective, the composition of the US of various states is irrelevant. Only the US as a whole is party to extradition agreements with other countries, since it’s a principle of international public law that the internal organization of a sovereign nation is a matter of that nation alone. So if the refugee were to be extradited from France to New York, but tried in Ohio for crimes not included in the grant of extradition, the USA would be violating the treaty with France.
There is a complicated extradition case currently being pursued in India.
Abu Salem was extradited from Portugal to India, on the condition that he not be given the death penalty (which exists in India) or more than 25 years in prison. The Indian government accepted this condition.
The case is now in the courts. However, the court is questioning the legality of such a pre-condition and whether it is binding on the court.