Problem solved!
And I've come to the conclusion that this is not an 10.11.2 El Capitan issue.
The essential problem was that Mail is out of sync with iCloud. When opening Mail it starts to download (sync), gets hung up at some number and then just stalls. In addition it keeps re-downloading emails that were perviously deleted. Quite Mail and relaunch did not solve this. Delete various Library files did not solve this. Here is what I did that finally worked.
Option 1
Open System Preferences, iCloud, and then Sign Out … but be careful. You will warned that doing so will delete files from you Mac for Contacts, Calendars, Reminders, Notes, etc. I only have these in iCloud so there was no problem deleting. However if you, for example, have any Notes 'On My Mac' they will be deleted. The solution in that case is to move everything over to iCloud. Once you're confident you can safely 'Sign Out'.
After signing out, sign in again. There will be a checkbox if you want to use iCloud for Mail, Contacts, Calendars, Notes etc. Leave it checked and complete the sign in – this means iCloud data will be sync'd to your Mac. Then the next time you first open Mail, Contacts, Calendars, Notes, etc the first thing that happens is syncing with iCloud. I did this with my wife's iMac (experiencing the same stalled download problem) and bingo … 'downloading' started, and completed! Previously deleted email did not return. All is now normal, and with super quick syncing among her devices.
Option 2
My MacBook was more challenging. In my case I had migrated most email folders (and about 2,500 emails) to iCloud in an attempt to solve the sync problem. After confirming they were in iCloud, I deleted them from 'On My Mac'. My theory was to put all my email+folders in iCloud only and it would stay in sync. This did not work – I was still stalling on 'downloading' and previously deleted email was reappearing.
So, I opened System Preferences, iCloud, and then Signed Out (making sure everything was safely in iCloud). I then Signed In with the box checked for Mail, Contacts, Calendars, Notes etc. Then opened Mail and it began to download … but this time completed! It took some time because it had to download 2,500 emails. It also does this in two stages, first it counts of the number of emails to be downloaded, after which it does the actual download (count + progress bar). When complete I was perfectly sync'd, which I confirmed by sending several test emails to myself and my wife. They all worked perfectly. I deleted, trashed … they were gone and did not reappear.
Once fixed, I moved my iCloud folders back to 'On My Mac', which I prefer for better security – email I want to keep is only on my Mac and not in iCloud. You can decide if this is important to you. When complete I deleted the folders from iCloud with no problem. I then did several more test emails and everything has returned to normal with super quick syncing among my devices.
Hope this helps.