Mail can't find emails in my "On My Mac".

Not exactly sure - perhaps someone else can chime in. I found this article that talks about it...

Mail for Mac: Rebuild mailboxes

Talks about re-downloading mail content, so might take a long time if you have a lot of mail.

Good luck...

I did a little research and V2 mail folder dates from Lion version of OS X (up thru Yosemite), V3 was for El Capitan and V4 is for Sierra (and newer?) -- see http://www.needhelp4mac.com/2015/10/troubleshooting-apple-mail-in-os-x-10-11-el- capitan/ for more info.

According to the post at How to transfer mailboxes from V4 (Sierra) to V3 (El Capitan) from a user named dianeoforegon, she says the V2, V3, V4 names are due to different versions of the SQLite databases. She sounds very knowledgable about mail on the Mac.

Doing a little bit more, it appears that searching in Mail sometimes doesn't give the expected results as it does not search the entire hierarchy of folders unless you choose the "All" button (top left of Mail Window) according to another post by dianeoforegon - see [Sierra] Really reindex mailbox?. You can also use Spotlight to search Mail by clicking on Spotlight on top menu in the top right of the screen. Apparently Mail uses Spotlight to do all it's searching so perhaps it might be more efficient to learn how to use Spotlight searching more?

Also see the article explaining searching mail at Mail for Mac: Search for messages in Mail

Just looked closely at your original post and the image of the Mail window - only the "Inbox" is selected and the "All" keyword - wonder if you selected all the folders on the left and also "All" that your search would work as expected? In other words the "All" keyword might only apply to the selected folders rather than all the mail? Seems to make a difference on my Sierra Mail setup - what happens if you try?

Good luck..

What exactly did you do to "manually transfer" them (via the "Import Mailboxes" menu command in File menu or ???)? What version of OS and Mail was the older iMac running? Any errors or odd things happen while doing the "import mailboxes"?

You didn't say what version of OS the old iMac was?

When you say that Apple Support asked you to "transfer the mail folder" you mean you literally copied the contents of the Mail folder from the iMac to your new MacBook, then fired up Mail on the MacBook and did it go thru some sort of "converting" step with a window informing of you it's progress or did it just all of a sudden show the mail inside the "On My Mac"

Also, didn't the old Mail folder already contain everything? What was different or the source of what you exported/imported from your "other mailboxes" you refer to that was not a part of the Mail folder?

On my system, I have one "account" thru iCloud and one "account" that was a result of an "import" via the "File/Import Mailboxes" menu command of an older Mountain Lion ~/Library/Mail/V2 folder. These two are physically shown as complex folder names in the ~/Library/V4 folder on Sierra (when looking at with Terminal) with long hex names as well as one folder named "MailData". The MailData folder contains several SQL databases and several XML files that I suspect is how Mail shows the various mail header item info like from, to, size, etc. So when you copied the contents of the old Mail folder on the old iMac to the new Mail folder on the new MacBook, there was no new database information created for these items that were copied. But the items you imported are searchable. Does that correspond to what you see?

Another possibility is that the two independent mail destinations (one is the "On My Mac" structure and the other is your "Mailboxes" structure) have independent databases and when you put something in the search box on the top right of the Mail window, it only searches one or the other, not both. But a little testing on my system (imported a 1.3GB mail folder from a Mountain Lion Mail folder) seems to negate that, as searching for messages with the Import folder selected yielded nothing for messages that I know are there but searching the larry. The only time the search finds them is if you select the folder that contains the message itself, which seems very disappointing as the computer is supposed to keep track of all that stuff and search everything I ask it to.

Well, son-of-a-gun -- after a while all my searches started working OK, so maybe I had not given the Sierra system long enough to build the databases for all the stuff I imported. Anyway now it finds everything I'm looking for the few tests that I tried. So never mind about my theories.

If you indeed did manually copy things without using Mail I wonder if you imported all that same stuff if your searches would start working?

If you do, make sure you have a backup of everything before experimenting. You might want to delete the stuff in the "On My Mac" structure (assuming that's the stuff that is not yielding any search results) before reimporting it. Make sure to do everything in Mail, not thru Finder or Terminal or other methods, since Mail has to keep track of everything it does and update databases and all that stuff as you add and remove things.

Good luck...ask more questions if this doesn't make sense...

I'm guessing you now have two of everything that was in the "On My Mac" or maybe the "Mailboxes" locations, as I'm still not 100% sure which one goes with the V2 and which goes with the V4 folder structures you have (I only have a V4 structure as I imported the old mail from a Yosemite system instead of move/copy the entire V2 structure like Apple Support had you do).

Perhaps one way to find out is look for a very unique mail message with some odd string of characters that should only be in one mail message, then do a Spotlight search for that string - bet you you have two (or more?) of them.

It's hard to compare the V2 and V4 contents due to different setups I had, but they appear very similar. One thing I read about comparing V2 with V3 versions of Mail was that the V3 storage requirements for the same set of mail was almost quadruple that of V2 (the example said that V2 took 17G to store the mail and the same set of mail in V3 took 60GB), so perhaps V4 has a similar issue? For details see the first link I mentioned above at http://www.needhelp4mac.com/2015/10/troubleshooting-apple-mail-in-os-x-10-11-el- capitan/ If you happen to talk to Apple Support again, perhaps they can give some clues/answers as to why the big size difference if you can't figure out the problem.

Wish there were some definitive document that detailed how the innards of Mail really worked on Mac so that all this made better sense and how to resolve unusual problems.

Good luck...

Congratulations -- looks like you're getting very close to getting things the way you wanted them to be.

How did you decide to remove all the attachments? Isn't that pretty important to keep around? Are they duplicated somewhere else?

Good luck...

Forgot to ask if you've tried the "Rebuild" option in the Mailbox menu (last in the list of commands)?