But today's you has diverged from yesterday's you just as much as tomorrow's copy would have diverged from today's original. At the moment of copying - assuming a sufficiently high fidelity - they would be practically identical. The differentiation would only happen over time.

So if you kill the original - maybe because the procedure is destructive or because it's a consciousness-saving emergency measure - the "copy" would have a pretty good claim to be a continuation of you.