Could I have moved something from my DIEP flap surgery 10 years ago that is causing this pain?

It is best to be seen and examined.  It is more common to have musculoskeletal problem than an issue with the flap.

The common practice for DIEP flap is to remove part of the rib to expose the blood vessels for microsurgical anastomoses. You may have pulled the chest muscles attached to ribs. Please see your surgeon o make sure that everything ok. 

I think you should go get evaluated, and likely get a chest Xray to check your rib alignment.  I imagine this would get better on its own, but when we do these surgeries, there is typically a small amount of rib resected near the sternum to expose the blood vessels we need to hook into.  So, it's not inconceivable something could have popped or strained.

If you have new onset pain even ten years out from surgery, then you are probably best to see your surgeon to be evaluated. 

You could have pulled a muscle or the pop was from scar tissue.  It would be best to have your doctor examine you.