I'm having the same issues. Can't put a finger on it, which is the problem. I've even restored the phone to factory settings, then re-installed from backup. I'm running 9.0.2 on a 6+ and the issue persists.
The phone is just not as snappy as it was with iOS 8. Every touch has a micro-delay that makes the phone feel very sluggish. I now play a game in which I press something then lift my finger as fast as possible to see how far away from the phone I can get until the phone responds to my press. I can get about 2 feet above the phone before Apple Music launches or the Settings icon springs up.
I've disabled various background apps and processes, to no avail. I will say that apps that rely on the Photos are much faster; Instagram and Snapseed used to take forever, I'm guessing because of iCloud Photos access, but now they're much faster doing whatever it is they need to do to display/edit photos.
But the OS itself is very, very sluggish. I have taught myself to pace my button presses. For example, after unlocking the phone with Touch ID, I wait about 2-3 seconds when the home screen appears; touch anything sooner and the input is ignored. Yes, I'm faster than the phone.