It almost looks like the border got "wrinkled" or "wadded" or pulled off-kilter somehow. Or that the quilter just put it on "auto-quilt" and walked away while that part was being stitched and the fabric shifted. It would bother me, too - a lot!
I was thinking that the puffiness would be able to "fold out and stitch down" - not from those pictures.
The only thing I can think of that might work - and would be a Problem In The Area - would be to unstitch all that quilting on the border - lay out the quilt on something flat - and then pleat out the fullness and stitch it down yourself - so it would look like a seam line when it it stitched - and have the borders requilted.