Going live after transferring WordPress

I’m 99% sure I know the answer to this, but want to double check. Here is the scenario:

We are switching hosts but keeping the same domain name (example.com). I successfully transferred the WordPress install (files + database) to the new host, where we’re operating under a temporary url.

When I define WP_HOME and WP_SITEURL as the temporary URL in the wp-config.php file, the site works as intended—no broken links or anything. http://temporaryurl.com/my-post-here works. When I REMOVE those two lines, all internal links render a 404 error. http://temporaryurl.com/my-post-here doesn’t work.

My question is, once I update example.com’s name servers to the new hosting account, will the 404 errors disappear automatically without me doing anything?