The protocol you want is SMB.
Windows has first-class support (obviously); but Samba gives Linux and BSD support that, in modern Desktop Environments, is exactly as good. Mobile devices don't tend to have OS-level support for it, but there are very good libraries to enable individual apps to speak the protocols (look at VLC's mobile apps.)
Even Apple has given up on their own file-sharing protocol (AFP) in favor of macOS machines just speaking SMB to one-another.
Yes, it's not workable over the public Internet. Neither is FTP, any more. If you've got a server somewhere far away, and want all your devices to put files on it, you're presumably versed with configuring servers, so go ahead and set up a WebDAV server on that box. Everything speaks that.