Why does noise-figure of components become less significant further down the chain in a cascaded network?
Think of the gains from the inputs of each stage to the output.  This is assuming a chain of amplifier stages.  The input of the first stage gets multiplied by the gain of all stages together.  The input of each further stage only by the gain of the remaining stages.

For example, you have a microphone amplifier with overall gain of 2000, where the first stage has a voltage gain of 10 and produces a low impedance output.  The signal out of that stage (and into the remaining stages) is now less susceptible to noise for two reasons:

The impedance is lower.  Capacitively coupled noise has high impedance because it's in series with a relatively small capacitor, usually well under 100 pF.  The impedance of the signal forms a voltage divider with this capacitor.  The lower the impedance of the signal, the more the same capacitively coupled noise is attenuated getting to that signal.

There is less gain from this point to the output.  In this example, the gain is 10x less than what the raw microphone signal undergoes.  This is because noise injected after the first stage isn't amplified by that stage.  In this example, noise after the first stage is only amplified by 200 to the output.