Online tool for receiving student files
One thing to consider is do you really need student assignments to be downloaded. I say this because I used to but I recently started using TurnItIn. However, I only use this for plagiarism detection, I do not use it for marking (university policy is that I MUST mark the paper copy...yes I know the dangers there, but that's for another day).

Since you didn't mention the larger issue you are trying to solve, I'll continue just with your direct question about how to download many student assignments without a million clicks.

What I did is simply setup an email account for students to email to (not my main email account). I instruct the students to use specific filenames (StudentID_StudentName_AssignmentName.docx) and they email me. I use a standalone email client (not webmail) so it all downloads to my computer. My client has a preview pane so I simply drag-drop to my assignment folder, move to the next message, drag-drop, repeat. I do this for 200-500 students per assignment per semester and find it does not lend itself to much more automation than this.

In all, it takes me 2 clicks per student (literally) and I end up with all student assignments in one folder. It handles ACK'ing the student on the server (auto-responder) but I could also use read-receipts but those don't usually work with webmail.

If you want to email students who did not submit, that's more complicated. I handle it during the marking process (let's see the file...ooops, they forgot...time to email them...or fail them for not submitting properly). TurnItIn will tell you which students did not submit.

As for students submitting multiple versions, as long as they submit with the proper naming convention, you will overwrite the older with the newer. I'm not sure if that is what you want or not but I would assume so.