You can drag and drop both
files and
folders into Opera, Firefox and Google Chrome that I know of. If you drag a
folder it looks a bit like you are browsing an FTP site. You can see the files and any subfolders in the folder as clickable links and browse potentially up to the root of the drive using "Up to higher level directory" links. If you click on a file link various things might happen depending on the file type (images, text files, videos open in the same tab, if the browser doesn't know what to do with the file, it'll suggest you "download" it.). What happens if you drag a
file depends on what kind of file it is also. Also you can specify a file or folder in the address bar using UNC path. Best to use foward slashes. As for other browsers, if you drag a
file into IE11 it tries to deal with it, and if you drag a
folder, it opens in Windows Explorer. Microsoft Edge seems to ignore drag-and-drop. Incidentally, someone told me their employer hastily rolled out Firefox ESR to them (they are on XP and not getting W10 until later this year) and... guess what? You can browse the C: drive, which you're not even supposed to see! He says he doesn't want to tell his IT helpdesk in case he gets in trouble for even trying it. So folder permissions may need attention if data confidentiality is important.
