Because the fare rules only list JFK-DXB-DEL as a valid routing to DEL

Changing it to LYS is unlikely to be viable given that EK has no fares from JFK-LYS

If you want to take the specific flights mentioned, you will probably have to change the JFK-DXB flight to the desired date and then purchase the DXB-LYS as a separate r/t and then continue to DEL

You would be far better off calling EK and asking them what the costs would be to do what you want to do. Looking at one of the economy fares, there would be a INR3500 change fee for the existing flight ( plus any fare difference should availability not be there in the current booking class )