They do. If they live in a country where the court has jurisdiction, they have obligations. (To show up, to cooperate, etc.) They may choose not to fulfil those obligations, in which case they can be punished (legitimately).

If there's no jurisdiction there's no obligation to start with.

Case in point: China has made it illegal worldwide to criticize what they're doing in Hong Kong[1]. That doesn't mean you have to listen to them if you're in a country that doesn't care.

[1]: https://www.axios.com/china-hong-kong-law-global-activism-ff...

I will disagree. Jurisdiction is in the eye of the beholder. Truth is that nobody has any obligations or anyone or anything, all the courts get their authority by the guns that back them up.

Wouldn't a US citizen still have those obligations if they, for example, had committed crimes in Germany but just happened to fly back to the US before being arrested?

Yes, but only because the US and Germany have an extradition treaty[0]. And even then, I assume the US government could still refuse to extradite for various reasons.

The US has no such treaty with Afghanistan, and is not a party to the ICC.

Whether or not the US should cooperate with the ICC is another matter, but the US has no legal or diplomatic obligation to do so. And in fact the US executive branch is required by US law to actively thwart ICC action against US servicemembers[1].

[0] https://en.wikipedia.org/wiki/List_of_United_States_extradit...

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

You misunderstood: the US government is the suspected "criminal" in this case.

If I, living in, say, the UK, criticize China's handling of Hong Kong, I am also a suspected criminal in their eyes. Do I have to care about that if I don't recognize their court's jurisdiction over me?

Criminals abrogate themselves from that responsibility by committing the crime. Their obligations are no longer relevant until the use of force is applied to make them fulfil that obligation - i.e. they are arrested and prosecuted. It is innocent civilians (i.e. non-criminals) who are fulfilling their responsibility to NOT commit crime - criminals are beyond that point.

America is getting away with its war crimes and crimes against humanity, simply because there isn't a system by which it can have its obligations not to murder millions of people enforced by external force. Yet. That day may change, one can only hope ..