1- You cannot change a ticket at the point of turnaround to return to a different country. This is an industry wide rule as far as I know.
2- Like Dave said, there are no JFK-LYS EK fares.
3- It will most likely be cheaper for you to return to DEL, then take a fresh ticket to LYS, than to add a stopover in DXB on your current ticket and get a rt ticket to LYS ex-DXB.
4- You could only do one of the following:
- Book a return JFK-LYS ticket on a different carrier, and pick up your inbound to DEL via DXB upon your return to JFK, Or
- Add a stopover in DXB on your current ticket, and book a return DXB-LYS ticket,
- Fly back to DEL as ticketed, and book a return DEL-DXB-LYS ticket.

As per pt.3, the last option might work out the cheapest.