The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
KITE, JET, EAGLE, and DRONE all fit the category "Things that fly". They spend time in the air. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Film crew roles
GAFFER / PRODUCER / DIRECTOR / EDITOR
GAFFER, PRODUCER, DIRECTOR, and EDITOR all fit the category "Film crew roles". They appear in movie credits. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things in a gym
MAT / BENCH / ROPE / KETTLEBELL
MAT, BENCH, ROPE, and KETTLEBELL all fit the category "Things in a gym". Training rooms are their habitat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Words after moon
WALK / LIGHT / STONE / BEAM
WALK, LIGHT, STONE, and BEAM all fit the category "Words after moon". Put one word before each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.