Is it possible that kidney stones can move out of your kidney without pain?

Yes. If they don't block the kidney, then you may not notice until it's in the toilet or on the way.
Kidney stone pain. "colic" from kidney stones is only related to the degree of obstruction - the only nerve endings in the lower urinary tract are "stretch" fibers: any stone(s) not associated with obstruction will probably not cause any pain!