Hi Simon,
When we connect an inductor in series with a capacitor we create a potential oscillator. When we drive it with a pulse the inductor and capacitor begin to oscillate. If it is an ideal inductor and ideal capacitor with no losses that oscillation would continue indefinitely even if we remove the pulse source.
This oscillation contains sine and cosine components as well as an exponential part. The sine and cosine are the oscillations we see. If we include a series resistance that provides some damping so if we remove the pulse source eventually the oscillation stops. The resistance controls the exponentially decreasing part so how much it still oscillates (if at all) and how fast it stops oscillating depends on how much resistance we add in series. For a small resistance it takes a longer time to stop the oscillations, for a larger resistance we may not see any oscillation or it will stop sooner.
In the MOSFET drive circuit we have inductance in the two leads that go to the gate and source, and a MOSFET input capacitance. This inductance and capacitance are in series and so can oscillate just like an oscillator. There is also a series resistance such as the lead resistance, but it is often very very low so we might see oscillations at the gate.
To minimize this oscillation the leads between driver and gate and source have to be very short. Also, if we use a drive transformer that transformer had best be wound bifilar.
Those solutions alone dont always reduce the ringing. To aid in damping this oscillation one solution is to provide a series resistance. This resistance is placed in series with the gate. It is often a fairly low value on the order of 5 to 25 ohms. This reduces the ringing because now there is plenty of damping, but it also increases the rise and fall times to some degree so there's a slight disadvantage. This increase however shows up as an advantage on the output of the MOSFET, where the slower rise and fall times generate less overshoot on the output.
You can try low values and see how much it affects the ringing and output overshoot.
If the leads are longer than a couple inches because they have to be that way, then a twisted pair cable should be used, and the same length going cable to all the MOSFETs is a good idea.