Extradition Treaties
Some thing people often forget not all extradition treaties are equal and some require strict adherence to the law and also require that the crime must also be illegal in the country being targeted by the extradition request. Part of that also being they must have adhered to local laws and using a US search warrant would be tossed out as a joke, the evidence would be tainted and extradition would fail.
So the law is a really bad idea, as it would promote bad investigation and failed extradition attempts in many countries.