If you're using the "Organize My Files" option in Mendeley, such that files are stored in another folder on your hard drive rather than in the Downloaded subdirectory of the Mendeley data directory, Zotero imports them as linked files rather than stored files, since that's the closest equivalent in Zotero (generally paired with the ZotFile extension to organize them for you in a similar way). But linked files don't sync, so you'd need to sync those another way.

If you want to keep them as linked files but move the folder (e.g., to Dropbox for syncing), you can set a Linked Attachment Base Directory in the Advanced → Files and Folders pane of the Zotero preferences, rename the folder, and then update the base directory setting. Note that this is only about the folder where your PDFs are, not the Zotero data directory.

If you want to use Zotero syncing instead (either WebDAV or Zotero File Storage, the latter letting you access the files on zotero.org), ZotFile might be able to convert the linked files back to stored files, but I'm not sure. If you don't use ZotFile, new files will be added as stored files by default.

(Since this is a bit different from Mendeley, I'll also note that, if you use stored files, you'd generally always access your files through Zotero rather than through your filesystem, since Zotero uses its own internal folder structure, though we have plans to implement a more human-friendly organization for stored files for people who do prefer to use the filesystem. Some people also use an OS smart folder that shows all PDFs below the 'storage' directory.)

I don't know off-hand if Mendeley moves files back to Downloaded if you disable that setting, and I would sort of be surprised if it did. But if it does, that would cause them to be imported as stored files, yes.

If you've synced all your files to Mendeley servers, you could probably disable that setting and use Help → Reset Mendeley Desktop, which will clear your local database and restore from the cloud. I believe that would then put the files back into Downloaded.