Tears technically and scientifically "run" or in the past "ran" down your cheeks or face, because they are liquid, liquid runs or flows.
Things that are a globe in shape like a ball, or a barrel, or wheels "roll" or a ball bearing inside a bearing rolls.
"tears running or rolling" as an expression means something that made you cry a lot, often used to mean with laughter.
Idioms are not technical or scientific. "tears run down your cheeks in the idiom", but it often gets misquoted to include my eyes, my face.