In broad terms, there are two approaches to returning suspects and convicted persons to the other countries where they are wanted, and would prefer for obvious reasons not to go.
One is what is called in the common law world ‘the backing of warrants’. Here, the authorities of the jurisdiction where the person is wanted issue their normal warrant of arrest, which is sent directly to the authorities of the jurisdiction where he is, who endorse it if it appears to be in order, and give it to their policemen to execute it as if it were their own. The suspect is then ‘lifted’, and handed over to the authorities of the country where he is wanted with the minimum of fuss—the unspoken premise being that the authorities of the requesting jurisdiction normally act lawfully and reasonably, and their arrest warrants can be taken at face value and acted upon unless a strong reason to the contrary can be given.