New domain initially will have no
SEO value itself.
If you point existing domain to the new one using 301 redirect, then SEs will pass all the
SEO value from old domain to new domain, and exclude old domain from indexes.
You can 301 each page (URL) individually, this makes a lot of sense if your internal pages rank well.
If they 301 new domain to the old one nothing will happen. In fact it tells SEs to disregard the new domain.