If there are big BGP changes, the world notices. If the BGP for the specific netblock your site is hosted on changes, the world probably does not notice. Your users will see slightly degraded performance (actually, in some cases it will improve), and (presumably) a number in the NSA will be <1% higher for a few minutes, but that's about it.

I've worked on software that literally (intentionally) publishes BGP updates every ~10 seconds. It didn't bother anyone, because it wasn't playing with many addresses.