Linking documents/pictures to other sites?

What is the correct way to use documents and images stored in a Home Site library on other SharePoint sites?

I need to embed images and create hyperlinks to documents on pages in other sites. I have been using the “Copy link” button, which creates a sharing link with individual user permissions.

The problem is that if I move the picture to another folder within the same library, the image sometimes disappears from the page. Occasionally the share link still works after moving the file, but not always.

What is the recommended way to do this, and is it fully supported?