For anyone that's arriving here super late, I actually figured out a little bit of a hack. However, unfortunately you need the paid software Keyboard Maestro in order to make it work. If you happen to have those, though, you're golden.
In my particular case, I wanted to immediately copy and paste a video into the app I was running as soon as it hit the Desktop (which in my case was because of the Screenshot app that comes with Mac OSX Mojave). The .mov file needed to be copied to my clipboard and pasted into the second program (called Ulysses).
1. With Hazel, set a rule so that whenever the file is added to Desktop, then show it in Finder. You need to select the extra option of 'Bring to Front' for the action.
2. In Keyboard Maestro, set the action 'Pause Until...', with the condition being triggered
when Finder is at the front. In my case, I set cmd+1 to both be the trigger for the macro AND the shortcut for starting a screen recording, so that whenever I went to record my screen the macro would be automatically triggered.
3. When the Finder is at the front, you should make the following actions occur:
1) cmd+C (as the file you've Shown is automatically selected),
2) wait 0.1s,
3) Activate Application [Ulysses],
4) wait 0.1s,
5) cmd+V.

This is how I automated copy and pasting of my .mov file into the Ulysses app specifically.