Does the magnitude of gene-expression changes decrease the more downstream a gene is from the origin of change?

Gene expression can be very well described by Hill functions, i.e.:

$$c_Y(c_X) = frac{c_X^n}{1+c_X^n}$$

when $X$ activates $Y$ and

$$c_Y(c_X) = frac{1}{1+c_X^n}$$

when $X$ represses $Y$ (omitting units and all sorts of constants for simplicity). For the common case that $n>1$, these functions look like this:

As you can see, they are far from linear, but sigmoids. For $n→∞$, these become perfect step functions.

As a result, gene regulation mostly works like a switch: As long as you are not close to the critical concentration ($1$ in the above example), changes in the regulator have a less than proportional effect. For instance in the above example, it doesn’t matter whether the concentration of your activator is $2$ or $10$, the protein production is pretty much $1$.

Therefore, “signals” in gene-regulation networks get stabilised instead of diluted. There are also some other self-regulation effects contributing to this such as genes that self-regulate, the availability of amino acids, and the limits of the translation apparatus.

Note that without these sigmoidal relationships, you would also have big influences of the inevitable fluctuations of the regulator. As a result, it would be impossible to keep the metabolism in some stable state – the only stable state in a linear system is death. Simply put, you would have to keep a very precise and regular diet, lest your body stops working.

Just to expand on @Wrzlprmft's great answer with a concrete example:

Sticking to your simple gene circuit with $X$, $Y$, and $Z$, now consider the possibility that $Y$ also activates its own expression via a positive feedback loop:

$X to underset{circlearrowright}Y to Z$

Here activation of $Y$ will be only minimally dependent on $X$. Once $Y$ is activated, the following self-activation makes $Y$ practically independent of further $X$ activity, causing $Y$ to drive its own activation until saturation.

If by decreasing $X$ to 50%, its levels are still above the minimum level required for activating $Y$, this would have practically no effect on the final (steady state) levels of $Y$. Conversely, if a 50% decrease brings $X$ below the $Y$ activation level, it would lead to a complete shutdown of $Y$.