My choice for the title of 'best military transport' would go to four aircraft.
First and foremost the ME 323 Gigant: The first 'roll-on-roll-off' cargo plane the ME 323 was very innovative and set the standard for military transports. The 323 directly led to other purpose built military transports like the C-123 (also initially designed as a glider) and the C-130. The Luftwaffe operated around 200 ME 323 with a combined payload capacity of a thousand Goons. It was a vital asset in the huge airlift of equipment across the Mediterranean supplying the Afrika Korps.
In the long range transport role I would place the following three aircraft as equal 'best planes' for their excellent range, payload and service to their countries.
The JU 290: Enabled direct communication between Germany and Japan via trans-Eurasian flights over the Soviet Union. Flying high and fast at night they were never intercepted by Soviet fighters. Was essential for transfer of technology, intelligence etc. Also had an innovative loading ramp design and was perhaps the best defended transport with turreted 20 mm cannons and machine guns.
A JU 290 at Stalingrad with its distinctive tail-lifting loading ramp deployed.
The C-54 Skymaster: With similar speed and range as the JU 290 the C-54 enabled effective trans-Atlantic flights between the U.S. and the U.K. Vital for fast transfer of important personnel and and critical equipment. An interesting anecdote is that the first batch of 200 desperately needed dive-flaps kits for the P-38 was transported to England by a C-54. Similar in appearance to a FW 200 it was unfortunately shot down by the RAF in a friendly fire incident, destroying its precious cargo.
The H8K: An excellent flying boat that was a vital communication asset for Japan. With excellent range and payload capacity it was the lifeline to the Japanese Empire's far flung outposts.
