To move your Box files to your new company's account, the best approach is often to transfer ownership of your folders to your new account. Simply invite your new account as a collaborator with editor permissions, then change their role to owner. For individual files or if ownership transfer isn't possible, you can collaborate with your new account and then move or copy the content. Downloading and uploading works for smaller amounts of data but loses version history. Consider third-party tools for more complex migrations.