Afraid that if you start rebooking yourself you put yourself at the mercy of the goodwill of the airline - your formal situation is then a no show for the next leg loosing all rights (I hope it was not a return ticket - if so, you may have lost your return flights too). They have to solve the misconnection/cancellation for you. Obviously too bad they did not want to book you on Emirates but such restrictions of rebooking to a non partner airline often play a role in such situations.

For a refund, at best they will pay the difference between RUN-MRU and your original ticket, which may be small.