We don’t have any special requirements for gallery backup and restore process. You can use buil-in backup module of the gallery or some other general backup plugin.
In the case if you have such situation like you described in your ticket it’s could happening because of the permissions problems.
Just check folder with images, make sure that it was uploaded to the new server and in the case if his folder and images exist try to check if you’ll have access to this images from browser. Just put to the link field your website domain with direct image folder and some gallery images on the end of the path. For example: https://link-of-your-website/folder/folder/images.png (something like this, of course replace all link elements by your original path and link)