Mail not arriving or long delays

Question

This is a strange one I can't explain! 
I am manually† migrating from a 2011 Mac Mini - to a 2010 i7 iMac - and have set up identical* mail accounts - however the i7 seems to have problems getting email from my Yahoo account (imap) in a timely manner.  There may even be emails that don't arrive at all in the Mail program - but certainly *SOME* (no rhyme or reason) emails do not arrive for hours, and in some cases a day or more.  By "some", I mean one or some emails may come through immediately, but other time adjacent ones might not.  Sender makes no difference AFAIK.
† (Manually Migrating = Reinstalling every program by hand, and copied user files - no migration assistant)
* (Identical = Every possible setting in Mail preferences - exact same OSX version - exact same Mail version - check for new messages every 5 minutes - junk prefs etc.)
As compared directly and in close proximity with - (All of which arrive quickly and reliably)
On the same i7 iMac -
Yahoo Messenger
Thunderbird
Checking the mail via website
On the 2011 Mac Mini -
Mail  (original/former main Mail program)
On Other Devices -
iPad 2
Android Phone
It's a bit disconcerting to see emails arrive correctly on all my other boxes and not see it come in (or arrive unacceptably late) on my target "new" desktop.
I like Apple Mail, but can't keep using it as it stands if it's going to be slow or unreliable.
Hoping for ideas - or suggestions on another Mail program (free).  I downloaded Thunderbird just because of this issue, but it's UI is way too PC like for me.... Although it's setup is easier and it has more features than Apple Mail - I'd prefer something a little slicker.

Answer 1

The only place I see the missing messages is on my iMac in mail. I use iCloud for mail on both my iPad and iPhone and I do look at it on iCloud.com.