What happens if the input of a voltage regulator gets out of the operating range?
What happens when the input voltage drops below the Output Voltage + dropout Voltage depends on the type of linear regulator it is.

In a MOS output LDO for example, the MOSFET behaves like a resistance below dropout as shown below (Source). So the output voltage drops fairly linearly according to the rdson of the MOSFET until it enters the off region.



NPN series linear reg's typically have a dropout of about 2V.

ADI have nice summary of the various types here.