Some force fields, such as "turbulence" can be moved or changed. But wind is one of those that does not seem to have any handles for controlling it with key frames. Its just on all the time, and the you can only change it and set its parameters before animating the scene. Is there a way to control it?
Most values displayed in the properties can be keyframed. This includes the Strength, Flow and Noise for Force Fields.
With the mouse over the field press
I or right click and select Insert Keyframe.
For more variation you can add a noise modifier to the value in the graph editor. Press N to show the properties sidebar to find the modifiers panel.