Using classic Outlook for Windows in business environments
I can not necessarily speak to any different third-party email applications since I personally don't use them deeply or work on their product teams. However, I can speak to Outlook. You should be able to continue to use any Outlook client that has multiple folders and mail in all of them and be able to sync quickly.
For any other third party client, IMAP is the main protocol and the standard that many email services use to sync.