These are the two most common actions to attempt correcting runout, and neither of them has any effect.

The first thing people do when they measure excessive runout and are unhappy with the measurement, is change the full length sizing die to a "precision" brand. Think about this for a moment. A die is precision cut on a lathe with a one piece reamer, the same type of cutter used to cut your chamber. The lathe can hold less than 0.001" tolerance. No die manufacturer has on their factory floor a lathe that is not holding that tolerance. This means that the die bodies from all manufacturers are essentially created equal. The only difference is the variation from SAAMI spec. For example, Lee Precision full length sizing dies are known to size to the minimum of SAAMI spec, which incidently makes Lee dies the best choice for semi-auto rifle reloading of range pick-up brass.

The die body alone cannot cause the axis of the case neck to deviate from the axis of the case body.

The second most common attempted remedy is to replace the reloading press with a "precision" model. Now think about this. The press doesn't size the brass. The press inserts into and removes the case from the sizing die. It's unreasonable to blame the dog for eating the goldfish when the cat is sitting next to the goldfish bowl. Any forming of the brass is happening in the die. The ram of the press may wobble, the shellplate may not be perfectly flush to the base, but this has no effect on the axis of the case neck in relation to the axis of the case body in the sizing die, therefore the press has nothing to do with runout.