Originally Posted by feline fanatic

Well you definitely want to put some kind of iron on interfacing on the knits to make them more stable. Like you would for a T-shirt quilt. It can be done but it is not fun to mix the two. I once did a memory quilt for a client that mixed knits and cottons and I wasn't too far in before I regretted taking on the job. I hated working with the knits, even with a stabilizer on them. Seams were a real pain when mixing the two. Now one thing that might work for you is to make the backing out of the knits (still with a fusible interfacing on them) and just sew large patches together, like 12" or bigger, much like
you would a T-shirt quilt. Then you can do more piecing on the top with the cotton dress shirts.