After an extended, initially unfruitful chat with APPLE and some troubleshooting, it was suggested that I delete a mailbox within Apple Mail, an Embarqmail account. I was reluctant to do this but decided to try it. I went to the webmail page for that Embarqail account after I deleted some emails from with Apple Mail. Those emails weren't removed from the webmail page. It therefore occurred to me that Apple Mail wasn't communicating with that Embarqmail account.
I then decided to delete the Embarqmail account in Apple Mail. I then re-entered the Embarqmail account as a mailbox in Apple Mail and restarted my computer.
When the computer came back up the Embarqmail account in Apple Mail began populating with the emails that existed on the webmail page for Embarqmail.
I then began deleting some Embarqmail emails from with Apple Mail. The proper movement of mail began with Apple Mail resolving the "movement" issue.
If you're encountering this problem tread carefully.