iPhone 5s + OSX Mail issue

Ever since I got my iPhone 5s, my Mail application in OSX has been acting odd. Used to be that when I get new emails, the Mail in OSX would get them almost instantly. Now after I got the iPhone 5s & set it up to get my email as well, the OSX Mail seems to be slow at getting new mail. Could even be minutes after I got it on the iPhone. And yes, I have it delete the mail from the system if I delete it from either source.

Today I checked my mail on my iPhone while at my computer. Said I had around 10 new emails. Right after I did that, the 10 emails popped into my OSX Mail. I'd prefer it if the Mail in OSX just was getting my email whenever new mail comes in, not when I refresh the Mail app on the iPhone.

I never had this issue on my old iPhone 4s.

Even now I have 1 new mail on the iPhone 5s, but it hasn't come through in the Mail on OSX. I've not changed any settings of Mail, but I did check the settings. They look fine, but mail is still arriving late.