Hi Penny,
So far, everyone's advice is focusing on the embroidery (which of course is important) but doesn't necessarily solve the problem of sagging. That is only corrected with quilting. Any block not quilted will sag.
My experience has been to free motion around the design itself and then the block as usual. The larger the embroidery design, the more you need to think of putting some quilting into the design. (Even large appliques sag)
In the quilt you have shown, I cannot see any quilting. Nothing is holding the fabric to the batting and backing.

HTH,
Keitha

Ok, now I am looking at my embroidered quilts wondering what is this dreaded sag? Can you please describe what you mean?
I just hoop a piece of stabilizer under my fabric an continue as I would with any embroidery. I do free motion quilt around the embroidery and it looks ok to my eyes. If I am doing something wrong I would like to fix it before I start to my next quilt.

Coffecozy:

Sagging will happen is there is no stitching to hold the batting and backing together. So, yes, the blocks need FMQ around the embroidery design or applique fabrics. Some of the designs have stippling already programed into the design steps. It is generally done after tack down of batting and background fabric.

If the design doesn't have the stippling, FMQ can be added. Some machines have stippling designs built into the machine. Easy to add to the applique design that doesn't have one.

Good luck!

Here is one example for EMB quilts. This was a multi-hooped block. Each design in the block had to be re-hooped. There were empty areas with no quilting in the original design. This pic shows after the embroidery. You can see the the wrinkles, etc.

The quilt was finished with a meaning around each design. When completed there were no sags in the quilt blocks.

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Thank you for the explanation. I guess that by always free motion quilting around my designs I have never encountered that problem. I will be sure not to skip that.