A stationary process' distribution does not change over time. An intuitive example: you flip a coin. 50% heads, regardless of whether you flip it today or tomorrow or next year.
A more complex example: by the efficient market hypothesis, excess stock returns should always fluctuate around zero. There is no trend; as soon as they can predict returns, traders exploit the trend until it vanishes. So no matter when you observed excess returns, it would still be distributed WN(0,$\sigma$).
As you said, it would randomly vary according to a white noise process.