importing images to new site

We are redoing our web site in a complete overhaul. Both are on WP. But we are using a new host. Development is being done on a .com url while the actual domain name is a .org. When done, the idea is to point the .com to the .org domain name.

We need to move about 1000 images attached to about 400 blog posts (along with each caption, title and alt title) to a new site. Using WordPress Importer even when I checked “everything” only the feature images were imported. The other images attached to the post appear on the new server, but are being linked to the current server, which when replaced, would mean they will all disappear. Is there a way to actually transfer the images from the current site to the new one without having to manually export and import each image and input the data all over again?

Thank you.