I don't usually see the periphery ring moving up and down. A record would have to be severely warped to resist the weight of the ring.
The one time I DO see the ring move up and down is when I play some old records that are ever so slightly smaller in diameter. In this case the ring doesn't sit ON the lip of the record at all points of the circumference of the lp. I play these records without the clamp, because a) it probably does more harm than good sonically, and b) i don't want the ring to drop during playback. Some records are in fact small enough to not hold the ring at all. Usually old records.
Check the underside of the ring, too (where it makes contact with the record). Make sure there's not some substance on it that's causing the wobble.
The one time I DO see the ring move up and down is when I play some old records that are ever so slightly smaller in diameter. In this case the ring doesn't sit ON the lip of the record at all points of the circumference of the lp. I play these records without the clamp, because a) it probably does more harm than good sonically, and b) i don't want the ring to drop during playback. Some records are in fact small enough to not hold the ring at all. Usually old records.
Check the underside of the ring, too (where it makes contact with the record). Make sure there's not some substance on it that's causing the wobble.