What kind of circuit is required to increase amperage with variable input voltage?
10 A is rather a lot in a car. It's = 12 V x 10 A = 120 W which will be the same as your headlamps.

What you are looking for is a voltage controlled constant current power supply.

Specifications:


Voltage: 12 V.
Control voltage: 0 to 10 V.
Current output: 0 to 10 A (or, if you can split the load, 2 x 0 - 5 A, etc.)


Alternatively you could use a PWM (pulse-width modulation) controller with the full 12 V switched rapidly (faster than the eye can discern) as shown in Figure 1.



Figure 1. PWM signal varying from 80% to 20% to 80% to 0.

Specification:


Voltage: 12 V.
Output current: 10 V.
PWM control: 0 - 10 V. (You could use 0 - 5 V if you divide down your control voltage with a pair of resistors.)