Ah. You must be using Gmail and Yahoo in "webmail" mode. I'm not sure what the advantages of that might be: I use Thunderbird and run Gmail (his 'n hers) as if they were IMAP accounts. It works very well. There is no delay that I can detect.
I have a selection of other e-mail addresses for various purposes (unlike Mrs Clinton), all handled through Thunderbird with no problems.
Yahoo I stopped using some years ago.