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.
WALK, STONE, BEAM, and LIGHT 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.
Category
Fast things
ARROW / FALCON / SPRINTER / COMET
ARROW, FALCON, SPRINTER, and COMET all fit the category "Fast things". Speed is the common thread. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Words before board
SCORE / KEY / DASH / SURF
SCORE, KEY, DASH, and SURF all fit the category "Words before board". Put one word after each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Baking staples
SUGAR / FLOUR / YEAST / BUTTER
SUGAR, FLOUR, YEAST, and BUTTER all fit the category "Baking staples". A bakery keeps these close. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.