This has been bugging me for a while, and I’ve never seen it explained quite right (in my opinion), so here’s my attempt.
Wonky borders, or dog-ear borders, are what you get frequently at the end of a long-arm quilting process (see Figure B). When you piece a quilt, and here in my very basic drawing I’ve just sewn a bunch of squares together (Figure A), all of the seams are locked together — except the ones on the outside! As quilters, we do not tie off our seams when we piece, so those end-seams are somewhat loose.
Figure A: The pieced Quilt. Figure B: What can happen on a quilt frame if the end seams are allowed to open
When you take the quilt on the longarm frame (here we are assuming there is no border), the top is generally forced to be square to the frame. Then, as the quilter quilts and advances the quilt down the frame, if the quilter is not exceedingly careful to keep all the stitching lines straight, those edge “open” seams will give a bit, and start to stretch downward (Figure B). If each seam stretches only slightly, say 1/32″ of an inch or so, but your quilt has small blocks (say 64 down a side), you could get 2″ of stretching before you’re done! As a result, you get the dog-ear effect shown in the picture. As a quilter, I use a 90-degree laser level to ensure that all of the seam lines are parallel as I quilt to prevent this from happening.
However, this is important to you as a piecer as well! Let’s say you’ve finished off your quilt, and you’re ready to throw on those borders. If you do like I used to do, and just whip those borders on, you may be in trouble. I always had a tendency to “pull” on the quilt a bit to make sure it was feeding straight into the machine. Guess what? I just opened up all of those edge seams, and now I’ve just created Figure B again, but now with a border that your friendly longarm quilter cannot adjust for.  That’s why it is so important to pin that first set of borders to prevent this from happening. You might even spray-starch your quilt ahead of time to make sure those seams are stable. It’s even more important for borders, because you can tug on these seams on all four sides, resulting in “wonky borders”.  The longer edge of the quilt still tends to get more distorted, just because of the number of seams involved.
So, here you go — why you can get wonky borders and how to avoid them!