You got a blueprint for a teleport system from Shingles. That should be the key to find the missing persons.
It says you should "combine large blast of energy with something moving 88-metres-per-hour to teleport it".
Unlike normal walkthroughs, in Nice Game Hints you can select just the puzzle you want hints for. Here are listed the puzzles for "Teleporter system" in The Darkside Detective - A Fumble in the Dark. The guide you choose only contains information about that puzzle, reducing possible spoilers for other puzzles.