Xen demonstrated this (for hypervisor-managed running processes) prior to 2005. It was one of the selling points of their virtualisation process, though it turned out that live-system migration had a few additional hiccups in it. Mostly works now.
If you're looking at back-end-mediated stuff, this is pretty much what happens when you synchronise browser sessions across devices, modulo the rate of transfer. The key is managing state intelligently.
This is also done, mostly, on server-side infrastructure, where front-end systems have little to no state on them -- individual client requests come through, state is managed usually in the datastore itself.
If you're looking at back-end-mediated stuff, this is pretty much what happens when you synchronise browser sessions across devices, modulo the rate of transfer. The key is managing state intelligently.
This is also done, mostly, on server-side infrastructure, where front-end systems have little to no state on them -- individual client requests come through, state is managed usually in the datastore itself.