I = E/R ?
Volts (across cathode resistor) = Current / Resistance
If current increases, voltage across the cathode resistor increases. If current decreases, voltage decreases. This is true unless you have the "negative resistance" of certain active circuits (where current increases even though volts pushing that current decreases).
Overall, a single-ended stage operated cleanly has similar current-change in both the positive & negative directions. The result is the
average current through the resistor is unchanged, giving a steady d.c. bias voltage.
You can get something else when the single-ended stage is not center-biased & is driven to distortion: one side of the wave flat-tops while the other side increases. That unequal-current in each direction causes the
average current through the cathode resistor to change, and the bias voltage follows.
However, when you drive the output tube towards max power, screen current rises. This is shown in the
top graph of Page 4 here by the way the dashed screen current lines increase at low plate voltage, which is where the late is pulled when it is moving towards peak plate current.
Or look at the Class A conditions and see that from idle to max-power the plate current only increases 15-20% (due to distortion) while screen current rises 50-280%.