On Apollo 15, 16, and 17 NASA filmed the Lunar Module taking off and leaving the moon. With no-one on the moon how could the the camera move to follow it and who brought the exposed film back to Earth?
It depends on which "film" and mission you are referring to. (The original question referred to "the first moon landing". Later edits refer to the last 3 landings.)
For the first moon landing, Apollo 11, the lift off was filmed with a motion picture camera inside of lunar module looking out the window. Obviously they carried that camera home with them and develop the film after returning to Earth. (The landing was filmed the same way: camera pointing through the window.)
For the last three missions, Apollo 15, 16, and 17, those were recorded from the TV camera on the lunar rover and used video transmission. No film and no processing was involved.
Elizabeth Howell — Universe Today 12/16/14 11:20am https://io9.gizmodo.com/how-nasa-captured-this-iconic-footage-of-apollo-17-leav-1671650186 provided the explanation also supplied by Uwe, above. It was a live TV/video feed from the LRV (lunar rover). Repeated on several missions.
Attached to the lunar module (ditched) NASA remotely controlled a TV camera routing feed through the crew module.
Could also use long range antennas.