IMHO, when you deal with crosses, or even crosses of crosses - you get mutts. Some people may be able to tell you what colors you will get, but you will never have a specific breed again. They're your birds and you can do whatever you want with them. I've crossed birds but I didn't think I could anticipate what I would get.
I crossed a Jaerhon and a Black Leghorn. the result was a beautify bird, er mutt.
I crossed a Black Penedesenca with a White Minorca. The result was a white bird with black specks that laid enormous eggs, even larger than both parent lines. It was still a mutt.
What you should get is more variability in all aspects, egg laying, size, maturity, feather quality, weight, feed conversion. So some birds might be better in some areas, worse in others or all. The more you mix, the less consistency you have.
Generally, over time, I would expect to loose thrifty birds... but I would expect it to take several years of crossing before one got really poor doing birds.
Full breeds should be pretty consistent in the above qualities.
When starting with two inbred pure lines it is fairly easy to know or at least guess what you will wind up with. The trouble begins when you take this first cross further and breed in more and basically unrelated blood lines.
As is the case with sex linked birds the first cross is going to be a known quality but as you go further and further afield with your breeding program more and more uncertainty will creep in, and finally this uncertainty will rush in and very likely overwhelm the casual chicken breeder.
Besides, if you are unwilling to cull a diseased bird then you will certainly be unwilling to cull those chickens with crooked toes, sideways leaning tails, defective keel bones, poor quality plumage, off colors, cross beaks, and a hundred other quirks and faults that will inevitability creep into any breeding program.
The place for cross bred chickens is to fill a niche like feather color sexing or some similar trait that can be expressed in the first generation cross. In game fowl crossing is only desirable if it creates a stronger, faster, more alert, better, or healthier, bird. Think of it like you were trying to breed a 6 Million Dollar Man with feathers. 99% of the time you will fail.
Mother Nature is a real hand full. In nature everything always tries to revert back to what ever it was before us humans got our hands on it.