It's a question audiences have been asking since 1933, so I thought this would be the perfect place for it :) Even Peter Jackson decided to side-step this particular issue in his remake.
How did they get "King Kong" back to New York?
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The 1933 movie doesn't show how King Kong was transported, but according to the script:
They possibly towed the raft with the ship, all the way to New York.
(Maybe Peter Jackson thought using a raft is a silly idea and therefore didn't include it in his version)
However, the raft idea was used in King Kong vs. Godzilla:
In the 1976 version they used an oil tanker to transport Kong: