Dubai terminal 2 to Terminal 3

They are quite far apart (opposite ends of the air field) so I would advice taking a taxi.

However despite what FlyDubai/Emirates website says, I found when I asked at check-in before FlyDubai will do it in which case when you arrive at T2 there is a transfer desk immediately to your left in the transit area after coming off the bus from the plane. Here Dnata will transfer you by shuttle to Concourse B at T3. From there you will have to go to the the right Emirates Transfer desk in whichever Concourse you are departing from. The Transfer Desk at T2 will tell you which one. The transfer takes about 20 minutes to get round the air field and but I think they only depart every 20 minutes. There is though plenty of time to do this with your 2hrs 20 mins..

Having done it he other way around too, it is much easier doing T2 to T3 than the other way around. If arriving in T3 the transfer desk for T2 is hard to find due to a lack of clear signposting. However Emirates and Dnata staff at Dubai in my opinion are very helpful and got me there.