Thank you all for your responses! This forum is awesome. But honestly, moving them to a different folder would probably just put an unnecessary strain on my computer.
I'm downloading dozens of video clips and images over the course of 4 hours (from our streaming service) in order to upload them to the (siloed) content management system (I know… my job is working on the workflow) and then I delete them at the end of the day.
Moving each of them to a different folder as I download them would only be helpful if I needed to keep them for future reference. So I'll just keep changing the download folder in chrome manually.