How can I migrate files from Box to Dropbox safely and quickly?

There isnt an official tool I'm aware of so the only thing I can think of is to download everything and reupload. 

If you can tinker a bit, a server‑to‑server transfer using the Box and Dropbox APIs is neat: a small Python/Node script can stream files from Box straight into Dropbox so nothing lands on your laptop, and you can preserve timestamps/metadata if you handle it. For company or large migrations, ask about Box Shuttle or Dropbox’s migration options — those are built for mapping permissions and big inventories. For low‑effort moves, you can use the web: create shared links in Box and use Dropbox’s “Save to Dropbox” (web button/extension) to grab individual files without full downloads. Some people also mount both clouds as drives to drag‑and‑drop, CloudMounter gets mentioned a lot in those conversations, which is handy for quick checks. Whatever you try, do a small test first, verify shared‑folder permissions and version history, and keep a backup of anything critical before moving everything.

Another option is a third-party transfer service, such as MultCloud, but you would need to pay for that service. Personally, I'd use either Cyberduck or Mountain Duck (similar to ClountMounter) to open the storage in each service and then copy the files myself.