Who is the issuaing carrier? I believe that is the one that you should contact but if you are on the earlier JFK-DXB flight then you may just want to talk to EK at JFK and have them get you on the later flight either the direct or if that flight is not available then perhaps the connecting flight via HAM. If you are on the late flight and miss it then you will most likely be going out the next day and you have the whole issue of who would be responsible for hotel etc and that should be the carrier that delayed you but if their is a drop of rain in the country they will claim weather so you have that hurdle to jump. Another pain could be that if you are on the late flight by the time you get to JFK there may not be anyone there to help you at midnight (I am not sure of the last flight out of JFK for DL).

If you flight is in jeopardy while in CVG I would talk to agents there as well as call EK and DL while in CVG if your flight gets delayed.