Have you sewn with Minky or Cuddle fabric? It's a poly plush fur fabric that comes in varying lengths and designs of furry goodness that's hard to resist. I've sewn with it a bit, and quilted it a lot. The key is pin, pin, pin, and if you aren't sure it won't stretch or slide, pin some more!
My favorite product is Shannon Fabrics' Cuddlesoft. It has a little stretch, but usually only cross-grain. I pin it to the frame with the selvages along the sides then I pin the cut edges at the top and bottom leaders with extra pins to control the stretch. I love quilting on this stuff - it really emphasizes the quilting and kids of all ages can hardly keep their hands off it's soft goodness!
Here are some tips for machine-stitching plush fur binding to a fuzzy creation.
Use a single layer for binding strips: 1-1/4" strips for a generous 1/4" finish. Cut strips from the lengthwise grain - same direction as the selvage edge. Join strips by overlapping short ends 1/4" using a zig-zag or mending (serpentine) stitch. Mine is #18
If the fur is especially long, I'll sometimes zig-zag the quilt edges after trimming to control the bulk.
1) Leave about an 8" tail and begin to stitch binding to the back of the quilt. I use my quilter's 1/4" seam foot with the edge guide. You may want to use a walking foot to be sure it doesn't stretch as you go.
2) Stitch to 1/4" from the edge, then backstitch a couple of stitches and remove from the machine; clip threads.
3) Fold binding back 45o to form the mitered corner...
4) Fold binding strip back down with the fold even with quilt edge; align raw edges together. Pin to secure.
5) Turn the quilt and begin stitching at the folded edge, repeating the miters at each corner. Stop about 8" from the beginning stitches.
6) Form miters on the corners of the quilt front; pin to secure.
7) Stitch edge of binding to the front of the quilt, pivoting at the corners. I really like the look of that serpentine stitch. Looks great on the back, too!
p.s. Keep a lint-roller, dustbuster, vaccuum, or garbage can handy - the fur really flies! After each cut, clean off your mat with a dry washcloth. Shake out the quilt over a garbage can, or just take it outside and shake it off!