Effects of power fluctuations on batteries?
A battery, over a short period of time, provides a relatively stable voltage. Call it V.
The power (call it P) that is consumed by a load is given by Watt's formula:
$$P = VI$$
where \$I\$ is the current.
If the instantaneous power is given by some formula such as
$$P = 10 + 2\sin (t)$$
Then the current from the battery (assuming the voltage constant) must be
$$I = \frac{10 + 2\sin (t)}{V}$$
Now, in fact, the battery voltage will vary a little under conditions of fluctuating current. A model that is often used for calculating such variations is to assume there is an internal resistance R inside the battery.
The terminal voltage \$V_{term}\$ then seen will be approximately
$$V_{term} \approx V - \frac{10 + 2\sin (t)}{V} R$$

how will the battery consumption or lifespan be affected?

For some definition of &quot;small&quot;, if the current ripple is &quot;small&quot;, the battery consumption and lifespan will not be affected.
It is hard to quantify where the ripple no longer can be considered &quot;small&quot;. Certainly if the battery has a rated maximum current, this should not be exceeded. However, one sign that battery lifespan or available watt-hours may be compromised is that the internal resistance R, seems to rise.
The reason why R will appear to rise is that R is mostly not &quot;real&quot; resistance, but rather the effect of the chemical reaction in the battery proceeding at a finite rate. Too much ripple, and the rate at which the battery's chemical reaction takes place will not be able to keep up. R will appear to increase.