popup says Mail needs to import my messages

The accounts are Comcast accounts and my iCloud account. I don't have gmail imported into Apple Mail.

Thanks for your explanation. I wonder why it took a week and a few days for Mail to get around to trying to implement the import to rebuild its database. Guess I will just have to suffer the download and then get redelete all the ones I have previously deleted. The thought is daunting but I don't want to keep logging in and logging out of the accounts.

I recall in the past that Mail attempted a couple of times to download every email again. I stopped the process but don't remember how. That is not possible in this case since I cannot access Mail at all now unless I allow it to download everything. Oh, well....

Mail successfully downloaded the messages from my accounts to rebuild the mailboxes and all looks ok. Just as you said, the messages that I had deleted from Mail had also been deleted from the Comcast servers so they were not downloaded. There were no duplications at all of undeleted messages. I was remembering what happened years ago and clearly the two circumstances were not the same. I am so relieved. Thank you again for explaining that this download was just Mail rebuilding its database.