I’ve done embroidery, with similar considerations.
I framed one piece in a shadow box; that’s the only one that has glass over it. It heightens the effect of the deep frame.
When I leave them in hoops, I stitch the back from side to side to maintain tension. The back looks like a giant asterisk * when done. I’ve also glued into a hoop, but didn’t like that as much.
Usually, I mount a piece on foamcore, sometimes with a layer of felt or quilt batting, and ladder stitch opposite sides on the back.