It's possible to extradite people only following an agreement about their treatment post-extradition. For example the receiving party could agree to not further extradite them. This kind of agreement happens fairly often. A common example is extraditions to the US from EU countries. Following a 1989 decision of the European Court of Human Rights [1], EU countries cannot extradite suspects to the US if they would face the death penalty in the US. Therefore in every extradition from the EU to the US involving a potentially capital crime, the US first signs a waiver agreeing not to seek the death penalty in the case.

[1] https://en.wikipedia.org/wiki/Soering_v_United_Kingdom

His terms probably include that the country promises not to agree to any extraditions unless the extraditing country also agrees to the terms of not sending him to the US and not allowing further extraditions without the same promise.

IE, Sweden would not give him to Australia unless Australia also agreed to not let him be extradited to the US or any country that does not promise to maintain the promise.