Push-pull happens when the trailer is blown slightly out of line from the tow vehicle, so it tugs or shoves on the hitch at an angle to the direction of travel. This in turn shoves the truck out of line, requiring correction from the driver.
This is why more aggressive sway control reduces/eliminates it. Sway control physically holds the trailer from swinging at the hitch. If the trailer can't get out of line, it can't tug/shove the tow vehicle.
Heavier-duty tow vehicles feel it less not only because of additional weight, but also due to their stiffer suspensions and tires. If there's less flex in the suspension and tires, it takes more force to shove the truck out of line.