To kwired’s point, a standard* VFD cannot “make” voltage above what is put into it, but it can deliver any voltage LOWER than its input. For the output, voltage and speed change together so at any speed lower than full speed, the output voltage is lower as well. To his example then at 1/2 speed, a VFD driving a 460V motor will only be putting out 230V, so if the input were at let’s say 410V, no problem at all, the VFD may not even know.
But if the speed command is set for full Hz, and the input is down to 410V, the VFD cannot put out any more than 410V. Lowering the voltage lowers the torque at the square of the voltage reduction, so if the VFD is still trying to operate at 60Hz with only 410V, the torque from the motor will drop to about 80% of rated (410/460=89%, then squared = 79.5%). If the load is light and/or the motor is over sized, it MIGHT run fine like that, but probably not. So what happens is that the slip increases, the motor draws more current trying to get back to slip speed, the drive takes notice and either trips on overload, or if programmed to, it will override the commander speed lower the output frequency in an attempt to shed load and keep running. So it that programming is selected, then yes, the motor will run slower if the line voltage drops.
* Sone newer drives are coming with an “Active Front End” for various reasons, and AFE drives CAN in fact boost the line voltage, usually by at least 10%.