While downloading isn’t directly supported by Popcorn Time, there is a workaround.
Go into the advanced settings.
Uncheck “Clear Tmp Folder after closing app” if it’s checked.
After you stream a movie or TV show, you should now find the downloaded copy that remains in your Popcorn Time cache folder. You can also change where Popcorn Time stores its cache files to give yourself easier access to the downloaded video files.
Be careful, as all those high-definition movie files can add up. Your Firestick may get bogged down pretty quickly if left to keep everything you watch online through the Popcorn Time app.