What happens if I move a website using SPAI from staging to production?

If you used ShortPixel Adaptive Images on a staging website and then you plan on moving that website to production then you shouldn't worry about the images, as SPAI should automatically change the URLs of the images and start serving them from the production website. If this doesn't happen, then make sure you flush whatever cache you may have.

There is 1 important thing to remember, though: you should associate your production domain to your account too: How can I associate a domain to my account?