TL;DR: Drag-and-drop doesn't work. Copy&paste does!
Before yesterday, my way of moving files between my local desktop and a remote one using MS Remote Desktop Connection involved a 3rd party service like Dropbox or email. Then, a kind soul took pity on me and showed me the proper way. In his words:
Drag and drop doesn't work but copy/paste does.
Simple, but veeerrryyy effective!
Note I've only tried this so far between 2 MS Windows machines, with some large-ish (30M) files.