Hand Work in Quilts Question
 

What could I do to stablize a cross stitch block that would keep the quilt washable? I've seen the quilts that are made with embroidery squares, which gave me the idea. How do those remain washable?

First test the thread to be sure it won't bleed. I would put fusible lightweight stabilizer on the back of the block to secure the stitches.

I use Pellon 911FF on the back of the completed square. There are lighter fusibles, I just like this one. Like Onebyone, it anchors the knots and cross threads.

I have one of my baby quilts that was embroidered by one grandmother and quilted by the other. That quilt withstood years of use when I was a baby and again with all 4 of my children. Yes, it shows signs of wear now. Some of the embroidery has faded, but most of the wear is in the fabrics used for sashing and border. My point is that I wouldn't worry too much about the embroidered squares. They are probably no more fragile than the rest of the quilt.

Pellon's Shape Flex is also a great fusible for the back of embroidery. It's woven, so more like fabric.

I've never seen any embroidered quilts that anything special has been done to the back. My thought would be that those knots and threads are nestled down into the batting. I do think perhaps these quilts are somewhat fragile but my concern has always been catching the top threads on something and pulling them rather than a problem with the back. I use an embroidered block the same as a pieced one.

is it cross stitch on mesh, or on fabric?

I cross stitch on Aida fabric. Here's a pic. Thanks for the help!

Embroidery which is usually on plain cotton fabric and cross stitch on Aida cloth are two different things. As Onebyone said, I would first test for colorfastness. If it's going to bleed, its going to be a problem washing it. I have no idea how Aida cloth washes or if it does. You might test a sample of that also.

do you have a small piece you don't like or will never finish? Make a batted and bound and quilted block and throw it in the wash. Your large piece is too gorgeous to waste on something that will probably wrinkle and bleed. Absolutely stunning.