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Not sure what you mean by “where the program is processing this documentary series.”
The “dirty paper” appears to be just that: dirty paper, although it’s possible that a program such as Photoshop was used to create the “dirty” look.
The multiple images in a single shot is achieved through the use of Picture-in-Picture, an FX that many editing programs can produce and which I suspect can also be produced in After Effects and Photoshop.