Is there a legal term for when one country is not capable of safely or effectively containing or treating a criminal, so they are transferred to another country to carry out the sentence? This doesn't need to be specific to America or any one country. I thought maybe I could find something in EU law, but no luck so far.
I'm not a lawyer, but I doubt it. The ability of a government to deal with criminals is kind of a bedrock assumption. That doesn't mean it's always true, but it does mean that the law (as far as I know) always acts as if it were true. The closest thing to what you're describing that I can think of is extradition--but that only applies if a criminal committed crimes or can somehow be indicted and tried in another country. It has to do with jurisdiction, not whether the country they're in now can effectively contain them. In fact, for extradition to even work, one country has to already have custody of the criminal and then hand him/her over to another country.
Well, if it is done without judicial process it is called "irregular rendition" or "extraordinary rendition." The US, for example, has allegedly done this several times when they wanted to torture someone for information. However, this would be an instance where the "arresting" state wanted to do something that its own law forbids, so they just take the suspect to a country where they can do what they want. But generally speaking, if it were a public issue, Robert is correct. It would be an extraordinary admission of weakness for a nation to ask another country to take care of its criminals. It also raises legal problems, because remember - a criminal is someone that has violated the laws of a state. So a murderer in, say, Venice, is a criminal IN ITALY. Because he has violated the laws OF ITALY. The US, for example, would have no legal right to detain the Italian citizen in question, because he has done nothing to violate US law. Extradition treaties, of course, give the US the obligation to detain this hypothetical murderer and return him to Italy, but that is the state of Italy allowing another sovereign temporary rights over one of its citizens, but only so Italy can then exercise its right as sovereign over the murderer in its own courts. If this were some kind of very extreme case, like if Italy had captured Magneto and the US was the only country that had the facilities to hold him prisoner, I would imagine that there would be a multilateral treaty drafted to allow the rendition of supervillains for storage in the American facility. You could also always have a situation where the criminal in question has committed a crime heinous enough that criminal codes seize automatic jurisdiction. In the US, for example, a person can be charged and tried for genocide regardless of where the attempt/act of genocide occurred, so long as the suspect is in the US. So all Italy would have to do to get our hypothetical genocidal maniac to be held in the US would be to put him on a plane to New York under guard, and let the FBI know that he would be landing at a certain time. See 18 U.S.C. 1091 for the law cited in this specific example. And be aware that generally, in this case the US would most likely offer extradition to the nation where the genocide occurred before initiating any criminal proceedings, unless it were a particularly high profile case.
Technically speaking, you can drag someone off to international waters and have your way with them, as you are not protected by any sovereign nation. That is if you don't want to drag them off to a Gulag like Gitmo or whatever "Black Site" you have to do your dirty work without getting too many questions. (The CIA supposedly has such Black Sites inside the states, but unless your in on that, we will never know for sure.) Or... If under Martial Law, you could have some bad stuff happen to you while under detention for crimes. (Like the Military has a spotless record in times of war in recent history. Abugraib?) Which you could be put in a prison of some sort and interrogated under the suspension of certain laws that would other wise protect you. If you really want to have fun, just have a secrete deal with NK and ship them over that way. They will get to go to one of the many lovely "labor camps" and be subject to some pretty terrible treatments as a result. Read some pretty scary shit about what goes on in those places, almost makes the Nazi Concentration camps look nice in comparison, which is kinda freaky. But as the others have said, they are very much correct in that without special and or extradition reasoning, would not send a prisoner to another country under normal circumstances.