Same problem. Gonna try disabling Wi-Fi Optimization
I noticed that too. And same goes for BBM. Sometimes I have to open up BBM again until I get my messages. I tried disabling Wi-fi optimization and seeing how that goes. It's under advanced settings when setting up your wifi connection. Hopefully this is the issue, since wi-fi optimization is intended to save battery life.
So turning off wifi optimization worked out for me and now my emails come promptly. The BBM issue was BBM itself. You might wanna do a reset afterwards.
Moto G xt1034 - currently stock kitkat with Google Experience Launcher
Kindle Fire HD 8.9 - CM11.0/SGT7
Atrix 4G - CM 10.1 by epinter using krystianp's kernels
HTC Status/ChaCha - CM7