Most DMMs have >10 M? input impedance. These 10 M? are usually always in series with the voltage input to the DMM.
If you use limiting diodes the maximum current flow through them will be ~100 µA at 1 kV input voltage.
High resolution DMMs use other approaches, which are more complicated. See for example the input circuitry of the HP 3478A multimeter (5.5 digits). It uses a multitude of measures against input overload, including two spark gaps, a NTC, voltage limiting diodes, active voltage protection (for the ohms current source) etc.
(Note: when looking at the 3478A input circuitry don't get confused by the front/rear panel switching. It's a bit involved, because there is now amps input on the rear. Also, the input hybrid is the same as for the 3457a, which has a few more features, connection-wise. So you can ignore the GUARD and precharge/guard amplifier. All the FETs in the hybrid are shown w/o gate ; the gate is controlled by a shift register internal to the hybrid. It's essentially a shift register + FETs + guard amp + trimmed resistors + some relay drivers.)