My question about sending internet packets through time was closed as "off-topic" as it was "non-mainstream-physics". This seems like mainstream physics to me, so it should fit on the site.
Your question about closure has already been answered under your physics question on the main site:
If you look at the first related question you'll see why time travel is basically considered impossible. On Physics SE we don't start with "let's pretend it is possible" because that's not physics, that's fantasy. What would happen is basically whatever the sci-fi writer wants to happen.
Your question is entirely unclear, because you don't say what "time travel" means. And once you specify what you mean (e.g. motion along a closed timelike curve), the answer is probably obvious--- under your assumptions (whatever they are) what happens is exactly what you've assumed will happen.