Most of these ideas rely on the fact that a 3rd party script does the "hard work" of creating a list of posts that match certain criteria. Then instead of showing them the usual way, use Justified Image Grid to display the result. This is done by passing the post ID list (e.g. 100,101,102) to a PHP template tag, that you can generate in the Shortcode Editor.
Replacing search results / paginated archive pages
For this you would modify your theme's search.php or archive.php file (preferably use a child theme). Find the loop and use it only to create a comma separated list of IDs. After the loop, insert the JIG template tag with the Post IDs setting receiving the variable with the list. Learn how to do this in the FAQ: Show search results with Justified Image Grid?
Non-paginated archive pages
Another way would be letting JIG do the "hard work" and fetch the content. You can use get_queried_object() to determine what the current archive is trying to display. It will have the taxonomy and the term ID as most important properties. What you need to do is pass these to a JIG template tag as the Filter by taxonomy and Taxonomy filter term settings. You'd insert this in place of the loop in archive.php - and get rid of pagination. You might want to use Load more or Infinite scroll in this case.
Related posts / posts by popularity
Find a plugin that works best for you to determine which posts are related. Or one that shows the most popular posts, in order. Don't care about the looks, just test the functionality. Once you are happy with that, you'd need to find where the plugin starts to display them, it'll have a loop somewhere. Hijack that and use it only to create a comma separated list of IDs. After the loop, insert the JIG template tag with the Post IDs setting receiving the variable with the list.
Opening just the content in the lightbox
Only use a specific type of page that you don't intend to be accessed directly, just in the lightbox. Like forms or help dialogs. In your theme or child theme you need to create a page template that has nothing but the content. Do not include header or footer, do not include any fancy layout, just what is necessary - a basic HTML skeleton. It should look pretty raw. On your pages, select this as "Template". In your JIG instance, add an image that has the JIG Link set to this page and JIG Target set to lightbox. Recent posts is not needed because you have custom links on the images. This'll make your post content load in the lightbox that supports iframes.