Is a liquid in a container always in equilibrium with its vapour?
The definition of vapor pressure means that for any temperature and pressure within the limits of a liquid system, there is a certain amount of vapor present above the liquid. The liquid and vapor at this point are in equilibrium because the vapor pressure does not change over time for that given temperature and pressure. When the liquid starts boiling, the system is undergoing a phase transition and in not at equilibrium because the system volume is changing and energy is continually being added to raise the system's temperature. Of course, we know the temperature does not start to change until vaporization has completed.

Vapor pressure of any given pure liquid is normally a strong function of temperature and a weak function of pressure. If you took a beaker of a pure liquid and put it inside a container that could support a vacuum, then as the pressure external to the beaker starts to decrease, more molecules from the liquid phase have enough energy to escape the liquid phase and exist in the vapor phase. If you continued developing more vacuum, the liquid would boil. This is seen on the phase diagram as moving straight down from somewhere in the liquid region until you reach the gas-liquid equilibrium line.