Hi everyone, I've been experiencing this setback too since I updated to Big Sur (I'm using a MacBook Pro Retina 13, early 2015). Before then, I used to click on a file and then, while keeping it clicked, swipe with three fingers of the other hand on the trackpad to quickly move it to the desired desktop.
But now, even swiping through the desktops takes more time than it used to. This is frustrating as it slows down the habitual workflow of the past 7-8 years. To make the process more fluid now I use three fingers to swipe up (Mission Control) and then move through the desktops, but not when I need to move a file.
None of the workarounds or fixes I read in this thread proved to be either permanent or long-lasting (I can employ one of them a number of times each hour, but that's just not practical), so I hope it is something that will be fixed with one of the future updates of Big Sur.