We see this issue with CardDAV (Contacts) Account from our MacMini Server, not CalDAV (Calendar) Account. The issue is simple, there is no longer a refresh feature in IOS 9 Contacts (Calendar still has one). In IOS 8 when you went you were in Contact application, then clicked on Groups, when you pulled down there was a refresh swirl and all was good; however in IOS 9 pulling down does NOTHING!!!!. The only time IOS 9 seems to update shared CardDAV contacts is when you either a new one, or edit or delete an existing one. To test we have an iPhone still on IOS 8, one on IOS 9, two iPads on IOS 9, one iPad on IOS 8, and two Mac’s still running Yosemite. When a new contact is created on IOS 9 device and saved it’s seen only on two IOS 8 devices, and Mac’s, not on the two other IOS 9 devices. Now edit a contact on another IOS 9 device and save, again the contact updates on both IOS 8 devices and Mac’s, not on the other two IOS 9 devices; however that new contact added on the 1st IOS 9 device now is on this IOS 9 device that you did the edit on.
Bottom line, Apple lost Contact refresh in IOS 9.
I can’t believe that just released IOS 9.0.2 still doesn’t have this fixed yet.