I have several third-party accounts set up. Currently when I open Apple Mail the activity monitor sometimes says downloading Messages and just waits for up to a few minutes for no apparent reason.
When I look at connection doctor everything is properly connected.
Question: Is there anyway to speed up this activity so that there is no waiting to download messages?
There are so many different factors at play here, including the mail servers, the internet, your computer, mail protocol, etc. About the only thing I can think of that you can control would be to “rebuild” the mailbox in Apple Mail and see if that helps.
It is likely that the problem is with one or more of your email service providers, but you need to do some tests to narrow down the problem.
First, rule out Mail app as the problem. Delete each email account in the Mail app, quit Mail, then re-boot (restart) your Mac. Launch Mail then add back your email accounts one by one. Assuming your email accounts are IMAP accounts, all of your messages will reappear as before in the appropriate inbox, sent mail, etc.
Now do a few tests - send yourself some email messages, from one account to another of your accounts, and maybe have your partner or a friend send a couple of emails.
Are your email messages still slow to receive? If so, the problem may be with your email service provider(s). My former email provider, Cox (a major U.S.cable TV and internet service provider), had notoriously bad (and slow) email service and repeatedly required re-entering passwords. Cox finally discontinued its email service, sending all of its email customers to Yahoo. I declined and set up new email accounts with Fastmail. This was a great choice - reliable, email messages load quickly, many good features such as email aliases and decent spam filters.
Using a gmail account in the Apple Mail app could also be a bit slow. Gmail accounts are not standard IMAP. They are similar to true IMAP accounts, but.there are some differences and performance (accessing servers, loading messages) is slower than standard IMAP accounts.