Impossible to reply accurately without seeing the situation but the probable answer is that you need to put in place page by page 301 redirects from the old site to the new site. The same would apply if you are referring to changed URLs on the same site. In both cases the pages don't need to be exactly the same, although reasonably on subject is helpful.
This allows Google to associate the old URL with the new equivalent, they will eventually drop the old one and in principle, transfer any value this holds. Any visitors still trying to access the old URLs will also be taken seamlessly to the new one.