"Yhey" may, but there's no reason for it.
If you have it set for push you'll receive the emails as they come in - the same as any properly written email app. The main differences (apart from the ability to handle Exchange Mail, which is non-standard [and Lotus Mail, if anyone still uses it, for the same reason] are the appearance, how you do things (tap the email, tap the little box next to it, etc.), how multiple accounts are handled (one of the things I like about
K-9 Mail and similar email apps is a unified inbox - all the emails in all folders from all accounts on one screen) - things like that. But POP3 is POP3, IMAP is IMAP and SMTP is SMTP. Any properly written email app should work with any properly written email server. The rest is personal preference. (You can even use Telnet to send and receive email - the protocols are among the simplest ones used on the internet.)
That's strange - that's exactly why I don't use the GMail app - deleted IMAP messaged being download again.