Just after start up and within a few meters of traveling, I hear a whine in the dash behind the steering wheel. Then the sound quickly diminishes to nothing. Its sounds like a gyro that comes to a stop. Other 2007 XKs must have this sound. What is it from?

 

Probably a dirty aspirator, a little fan behind those little slot openings on the lower dash.
Lots of posts on it if you search.
Here’s a short and to the point thread;
https://www.jaguarforums.com/forum/xk-xkr-x150-33/aspirator-fan-noisy-265684/

 

After a cold start, mine will make a slight whirring noise originating in the center of the dash. I believe it waits until sufficient heat is coming from the engine to warm the cabin at which point the dash internals start moving shutters and diverters to route the air to the correct location. After a few seconds, all sounds cease with the delivery of warm air. I have not heard it when the system is in cooling mode but will listen for it when the season arrives. Good luck!

 

Try this site he shows you how to pull it apart and clean it.

 

THIRTY MINUTE VIDEO???? Ugh.

 

I watched the whole thing - too bad it was like those unbox videos. savoring every moment.

At least we get to see the whole aspirator module.