[bdamm]

I think this is actually not the case. A Boeing 737 totally empty and in the smallest possible configuration is 61,864lbs. If 20 people move up to 1st class that is less than 7% of the overall weight. With the engines and fuel in the centerline, that represents even less in terms of overall torque change on the center of gravity. The aircraft would adjust for this with the tiniest of movements on the elevator, so small it very well might be in the hundreds of an inch.

It is not about balance.

[pintxo]

Here is an example for an A320neo where LH seems to have resolved the issue by not booking the last row due to GoC issues [1]. So 6 people can make a difference.

[1] https://simpleflying.com/lufthansa-a320-cog-economy/

[technicaldonut]

I think this depends on how full the plane is.

Moving a few people probably doesn't cause an issue if the rest of the plane is completely empty.

If you move 20 people from the right side to the left side of the plane this will probably have some kind of effect.

On smaller regional aircraft they sometimes ask people to move seats to the other side or back to front.

[bdamm]

Moving people left and right has absolutely zero effect on the controlability of the plane, because the ailerons can compensate for much larger movements than that. What it might have an effect on, though, is the overall efficiency with which the plane flies. It is more efficient to have the center of lift and the center of gravity matched, than not.