Collapsible luggage?

It doesn't need to be a rolling duffle, does it? We routinely take an empty plain ol' duffle bag in my carryon, that gets filled up and checked on the way back. I used an old military duffle once. Or, alternately, buy one over there. Plenty of cheap luggage dealers just about everywhere.

There are all-fabric (nylon?) bags that zip closed and sort of fold into their own pocket. They are, in effect, duffle bags with a rectangular rather than round shape. I've bought them in various weights. Folded up, the larger, heavier-fabric ones are about the size of a paperback book and will hold quite a lot; I have one that's probably close to the maximum size for a European-flight carry-on. The thin-fabric ones fold up smaller but usually have a lower capacity. The latter weigh almost nothing but are not sturdy enough to carry heavy purchases throughout a trip.

What I'm thinking of has no frame or padding to protect the contents, so you couldn't use it for breakables if you wanted to check the bag. I'm not sure the fabric would hold up to being checked even if the bag contained only soft items; it would be a risk.

I make sure I have several of the fold-up bags on hand in different sizes/weights; I pick them up when I see something inexpensive and useful in a store. If you want to buy in person rather than online, look in places like the Totes outlets in malls (where you'll probably find something under $10), the Container Store (may be pricey), on the travel-gear rack in stores like Stein Mart, etc. You might also check Walmart and Target.

You must be looking for something like a light weight pack-able duffle bag with no wheels. Something like 44 liter and 6 oz.

Once in the Naples airport I was told I had to check my camera equipment, and I didn't have enough room in my bag. I did, however, have a zippered tote I was using as my "personal item" for the plane. I put my camera stuff in it, ran downstairs and checked the bag. I was a bit nervous about packing lenses etc. in a soft-sided tote, but it all worked out.

I now carry a smaller zippered tote that folds into itself, and could be used as an extra bag if necessary. And my DH carries a smaller backpack that fits easily into his regular traveling pack. It could be checked, if necessary.

The bag Edgar linked to is sort of a Cadillac version of what I have. Mine cost between $1 (ages ago, and very flimsy) and about $8, I think.

Excellent suggestions all around, thanks!
I'm thinking that I will transfer clothing/unbreakable items in packing cubes into one of the nylon/soft sided duffles recommended above and then put the bulky/breakable items I am picking up in my hard side spinner.
After looking at all of the above suggestions and a few options on line, it seems the criteria I should look for in this magical bag would be: strong enough to withstand checking, lightweight & compact, but with straps that will hold up to upper weight limits of a checked bag (yet not flap around and get destroyed by luggage conveyers).
We travel to Europe for extended periods often and I foresee us needing such a magical bag for future trips as well. Because of this I'm willing to invest in something substantial if it meets my previously mentioned criteria.
Anything else I should add to the criteria? Any specific brands that could be recommended? So far I've looked at L.L. Bean, Bago & Biaggi.
Thanks again for all the great advice so far!

Or you could carry the lightweight/flimsy duffle, fill it with used clothes to carry on the flight home (keeping within the weight limit for carry-on), and put purchases in your sturdier checked bag in the space the dirty clothes took. (Dirty clothes also make excellent padding.) I have a fold-up duffle that cost about $5 at the local (upscale suburban) thrift store.

I used a basic Jansport duffle for years, currently an REI bag. No need to get high tech. A removable shoulder strap helps with sending through baggage system. I sometimes duct tape the handles together, and use nylon zip-ties to keep it zipped. As Laura suggested mostly dirty clothes go in the checked duffle.

Here are a couple to consider:

Eagle Creek Packable Duffel

Travelon Packable Bag

Kate, I use a Nike foldable duffle, big enough but not too big, strong enough but not too heavy. Near the end of my trip, as I prepare for my return flight, I put non-breakable things in it, like dirty or clean clothing, shoes, books, etc. and check it. Then I put anything more fragile or valuable in my carry-on. Never a problem. When I have had any liquids like wine or olive oil that I could not carry on, I have found an empty smallish cardboard box, usually among many thrown out by stores for pick-up in the evening, and used that, with the bottle(s) wrapped in plastic and clothes, then inserted in the box, again surrounded by clothes, in the duffle so nothing moves around in the duffle. Still, never a problem. I used to have a similar Samsonite folding duffle, which I bought cheap at a Samsonite outlet store, but I found it was much too large, and while it worked, I could not fill it enough to assure that such a box with wine might not flop around and get broken.

I also would switch out the carryon for the flimsier packable bag, check the carryon and put the packable bag in the overhead.

We are in Ukraine today with a RS convertiable back pack for me, a small truly carry on legal hard bag for her and an Eddie Bauer duffel that folds into its own pocket and measures about 10 x10 x 3/8 inch when folded and 10 x 10 x 24 inch when opened for expansion when she goes shopping. Eventually all the precious things will be in the hard case, clean clothing in the RS bag and dirty in the Eddie Bauer.

This Sea To Summit Ultra Sil 40 liter duffel goes with me on every trip:

http://www.seatosummit.com/product/?item=Ultra-Sil+Duffle+Bag+&o1=0&o2=0&o3=324-36

https://www.amazon.com/Sea-Summit-Ultra-Sil-Duffle-40-Liter/dp/B006CG2ZEA

It weighs less than 3 ounces and stuffs into a sac about the size of a baseball. I often check my main backpack on the way home and use the duffel as a carry on. So far, it has held up well over multiple trips to Europe.

The only problem is that it's insanely expensive for what it is -- about $45.

We have used a Rick Steves Hide-Away Tote for this purpose. It is advertised as 22x16x8. 4.8 oz weight. The fabric feels substantial for something like this. You could use a Travelon Bag Bungee to secure it to the handles of a rolling bag while moving between destinations. Or on our last trip I used a sleeping bag bungee cord to mimic a bag bungee. When not in use, it fits easily in one of the outside pockets of my bag (Rick Steves rolling carry-on).

A simple, cheap cinch sack may work. Check out the camping goods section at Wal-Mart or a laundry bag from Bed, Bath, and Beyond. Of course, you can always go nicer. I like to buy nice stuff from time to time to support the economy. It 's a sacrifice for my fellow Americans and their businesses, but someone has to circulate some funds.

@barbnrob, your country appreciates your sacrifice.