Get yourself a car with the biggest, mostpowerful engine you can find, then feed it through the narrowest tubing you can find.
Come back and let me know if an inadequate supply can affect the performance.
If voltage drop is excessive enough that the motor is not going to accelerate to full speed, current will be high, voltage drop will be even worse, something is going to give and it will shut down or burn up.
If connected to a variable speed drive then we have a different situation - starting current will be low, overloading will start to happen somewhere before full rated speed is reached. The transformer will attempt to deliver the load that is demanded but will have increased heating as a result. If transformer is not able to deliver power needed voltage will drop, current will go up - making voltage drop even worse - something will either respond and shut it down or something will eventually burn up.
In regards to thoughts about a loaded motor- If a loaded motor is able to start then there is enough capacity for it to run.
If Billy Idol or John Denver is on your Ipod go and re-evaluate your life.
If the motor is connected direct on line you have this sort of characteristic:
Clearly, the highest current is at starting. Thus is a double whammy for the supply voltage. Not only is it the highest current but it is at poor power factor which results in greater voltage regulation than the same current at good power factor. Then you're also hit about the lowest torque per amp. If the motor can get past that, things get better all round, so stalling isn't like to be caused by transformer capacity.
This is just confirming what others have already pointed out.
If the motor is fed by a variable frequency drive (not mentioned in your original post) and the supply voltage drops then so will the DC link voltage. Most VFDs default to fixed V/f ratio so there at at least a couple of possible outcomes. Depending on how low the voltage is I can see a couple of possible outcomes. The low DC link might prevent the motor getting maximum frequency thus limiting its speed. And if it's lower than the drive settings the drive will trip. Again, the motor stalling isn't a likely outcome.