The dash cluster isn't grounded really well. Depending on year model you may also notice if you hit your signals while blasting down the highway, the speedometer will momentarily drop back.
Dash cluster ground is at the metal tab under the dash where the diag connector is attached. Single black wire with a ring terminal. You could try cleaning that metal tab up and be sure the screw for the terminal is nice and tight.
Unless you've got something really out of the ordinary going on, you're not really seeing a significant voltage drop. You can confirm this by metering at the battery while having a helper the signals. Sure there will be a minor fluctuation but not much. You're just seeing an exaggerated symptom of that at the cluster because of the marginal ground.
Richard