I have a mate who inquired about making the switch when he started from Airbus to 777 and he was denied. Perhaps it has happened - I don't know. That said, keep in mind that most of the growth going forward is on the Airbus side (A380s and eventual A350s). He actually likes the A330 flying (despite a lot of night flights to India) - the variety of routes is very interesting for him as he gets to see parts of Africa and the ME he would never see otherwise (something he values - maybe not other people). Good or bad depending on your perspective, you could be an A380 FO in 2-3 years from the start. This could be your view:
Photos: Airbus A380-861 Aircraft Pictures | Airliners.net
If you are concerned about not getting home to the States as much on the Airbus fleet, I hear the A380s will start to replace many of the 777s on US routes in the coming years (that's the rumour).
I would say go ahead and make the request for a change to the 777 but set your expectations accordingly - don't be surprised if they say no. With all of the desparate pilots applying at EK - the company definitely have the LEVERAGE.
Make sure you go to EK with your eyes wide open regarding the negatives of the job - just read the many perspectives (positive and negative) on these Pprune threads and understand it is not like flying in the States. You would be a guest in their country and play by THEIR CHANGING RULES.
Let us know if your request is granted (just curious - you can PM me). Good luck!