The valve characteristic tends to linearize the response to a change in the process. For instance, if the gain if the process is linear, a 10% change can be made with a 10% valve trim movement, over the entire range of your process, then a linear trim valve will be easiest to tune, and can be made most responsive over the entire range.
If the gain of your process is non-linear, then a valve with trim cut to compensate for that non-linearity can give you a more stable process with equal response.
One example I like to use is the pressure reduction at the output of a centrifugal pump as the flow increases. So if you are controlling flow over a wide range, a very small change of the valve Cv at the first 10% of the valve opening produces a large flow change, because the pump pressure is very high.
As the valve opens, the flow increases, and the pressure head developed by the pump decreases. So to get an equivalent change in flow for a change in valve opening, the valve must open more and more, to compensate for the lower pressure at the inlet of the valve. Of course, this assumes that the valve discharge pressure is essentially constant.
So to apply the valve properly, you must look at the pressure drop across the valve trim at all valid operating points in the process. Then select the trim to linearize the response. Fisher Valve Control Book has very good examples of generally what sort of trim to apply to what processes.
A very common error is the application of equal percentage trim on a process with a linear response. If your centrifugal pump is operating in a fairly linear portion of the pump curve, you do not want to apply an equal percentage trim valve, because you will have to tune the control loop for the portion of the process which exhibits the highest gain. This usually gives you sluggish response at the other end, penalizing your plant and manufacturing operations.