Also, do you cut most of your borders lengthwise, parallel to the selvedge? Fabric cut in this direction stretches less.
You obviously are not pulling on the quilt body or borders as you attach them. You are also blessed with a machine that feeds well. After you attach the borders, lay the quilt out on a flat surface. If it's wavy, you can always remove & fix. Quilting should be fun. If you are happy with your results, then we certainly are!