It is possible to migrate the state of a distributed system by designing a protocol such that all of the components stop sending information around, so that you can be sure you did not miss anything. But I think that violates the spirit of the phrase "running state", since you essentially stopped the system. It's also not necessarily so that post-migration, the system will act the same.
To be more precise: any "possible outcome" of the snapshot was also possible at some point during the original execution. And likewise, any "impossible outcome" of the snapshot was also ruled out at some point in the original. So if viewed from the outside, the fact that you took and restored a snapshot is indistinguishable from the normal situation of having an asynchronous network. It can't cause the system to exhibit any behaviors that weren't otherwise allowable.
(Apologies if I'm just restating what you already know; other readers may not be aware of it. This might be the same property that you're calling "integrity".)
This is reminiscent of one of Scott Aaronson's comments from the original article. If quantum effects are significant in the brain, then it's impossible -- even in principle -- to "perfectly" copy it such that the original and the copy are guaranteed to behave identically. But if the original and the copy are statistically indistinguishable, it amounts to the same thing.
And even the Xen example is rough; the new copy of the system will be mostly functionally identical, but any reliance on the state of external things (like remote storage or IO or the state of unfair nondeterministic locks) will rapidly cause state to diverge.
Which only begs the question about how high-fidelity a copy has to be to be a "copy", even in the case of a hypothetical consciousness copy, and leads into the even more subtle question of the time evolution of a person and whether the me of now and the me of twenty seconds ago can be said to represent the same singular "consciousness" at all.
That's a feature of the Xen hypervisor.[1] You can even migrate a process from one machine to another without stopping it.
[1] http://wiki.prgmr.com/mediawiki/index.php/Chapter_9:_Xen_Mig...