Do they? It depends very much on what aspect of performance you want to optimize.Meter movements like to be driven with a current source.
A galvanometer is a second-order resonant system. There is a particular source resistance that results in no overshoot. The system has then a damping factor of 1. It makes the response a little slow. Making the system a little less damped results in some overshoot and oscillations that are slowly decreasing in amplitude (damped oscillations). By carefully tuning the damping factor, it is possible to increase the apparent response speed with negligible oscillations.
Current drive (infinite source impedance=>undamped system) produces a large overshoot and long oscillations, which is not a desirable effect, particularly for measuring quicly varying signals.
It is not the intention. The purpose of resistive drive is to optimize the ballistics of the system.Adding a series resistor with an op amp voltage source sort of approximates this.
The VUmeter standard thoroughly describes the target response and how to achieve it.
Which would result in a hardly usable system.Alternatively you can use a transconductance amplifier to create a current source output from a voltage source input.