E.g. You spawn a process on another computer and then the connection drops.
In practice it might not be an issue. The reason you'd typically do something like move processes across a WAN is because you want it to operate next to data it is making heavy use of. The copy that booted up local to the data would continue operating, while the copy at the point of origin would suddenly see the data source go offline.
Now of course more complex schemes can be devised, like if the data source is replicated and so both copies continue operating. Maybe a metric could be devised for detecting these instances when the partition is healed, and one or both processes are suspended for manual resolution? Or maybe programs just have to be written with the expectation that their capabilities might suddenly become invalid at any time, because the capability sides with the partition that includes the resource? Or maybe go down the route of making the entire system transactional, so that partition healing can occur, and only throw away transaction deltas once receipts are received for all nodes ratcheting state forward?
It'd be an interesting research area for sure.
Oh, and anytime as needed the OS could serialize out your process, pipe it across the network to another machine, and resume. Useful for load balancing, or relocating a program to be near the data it is accessing. Unless your program pays special attention to the clock, it wouldn't notice.
I still think about Amoeba from time to time, and imagine what could have been if we had gone down that route instead.