I've experienced this effect many times with analog meters.
It's not directly dependent on the current sensitivity of the movement, since the 'interfering force' is attraction between the (effectively) grounded metal needle, and charged areas on the meter cover plastic. But the charge-induced offset will be inversely proportional to the helical spring elasticity modulus. Since more sensitive meters tend to have 'lighter' springs, they do tend to deflect more electrostatically.
It happens more with plastic-faced meters than with glass, because plastics tend to transfer charge to other materials rubbed across them, more than glass does. But glass does too.
Once there's a charge on the meter face surfaces (inside or outside) the needle will also move as you wave a hand/finger around nearby, since your wet-salt-bag body is shifting around the electrostatic field lines originating from that electrostatically charged plastic area. Thus varying the force on the needle.
Generally such charge accumulations are on the surfaces of the plastic, and can be easily removed with a breath of moist air, or anti-static coating. But I've always wondered - like any good insulator, plastic can hold internal charge concentrations, that are trapped and no amount of surface wiping will eliminate them. A good conductive coating will shield the electrostatic field originating from an internal charge, but lacking that... I wouldn't be surprised if internal trapped charges result in permanent non-linearity errors in some analog meters.
You can see evidence of internal trapped charge in the dust patterns that accumulate over time on the insides of some injection-molded plastic cases. Especially when the plastic is white. As the hot plastic is squeezed through the passages of the molding machine and into the mold, the flowing plastic acquires volumes of electrostatic charge. Which become trapped in the cooling molded item. Then dust is attracted by the permanently present electrostatic fields, and shows up as dusty 'flow lines' on the item surface. You can see how they reveal plastic flow radiating from the injection molding points.