The whole idea of "designer breeds" just seems like a marketing ploy to me, but there are breeds that are cross-bred to improve their working ability. I'm thinking of one very old example - the Alaskan Husky! There is no such breed, an Alaskan Husky is simply a cross-bred dog bred for stamina & speed, and used for pulling sleds. While they often breed with a husky, some Alaskans don't have any husky in them at all.
I guess I didn't realize that a pug-chi was a "designer breed" - I just thought our little friend Lucy was 1/2 pug and 1/2 chi. I'm a little behind the times.