You cannot walk between T1 & T3. They are completely separate terminals. Emirates fly from T3 (Concourses A, B & C). The other airlines fly from T1 (Concourse D). T1 used to be Concourse C, but that changed a few years ago when Concourse D opened. See the satellite image below. On the left is T1, on the right is T3. There's no bridge as it's an active taxiway between the terminal.




"Terminal 1’s D-Gates are reachable via a short trip on an airport train. In Terminal 3, passengers can walk between B-Gates and C-Gates and take a short airport train to A-Gates."

This means that you take a train from Concourse D from the T1 check-in area. That's the only the way to get there (other than via the airside bus). You can see the train on the above image.

So getting to T3 Departures should be no problem.

No problem getting to T3 departures, but you then can't get to T1 departures without backtracking and taking the bus. The bus a long time, and you'll then need to clear security again on arrival at T1.