It’s possible to free range several varieties of chickens and keep exactly one variety pure. The rest will be mixes. You control which variety is pure in the barnyard by controlling what the rooster is.If I am just looking for a pet. I really don’t care what breed of dog it is. If it catches my eye. Seems to have enough between the ears to learn basic commands. With good temperament and character. I will take a chance on it. If I am looking for a dog to perform a specific duty. I.E. Livestock guardian, herding, hunting, etc. I am going to seek out a dog bred specifically for the task. From a reputable breeder, that uses their dogs for such activities. As for chickens, it really erks me. When someone advertises they have a specific breed chickens. Then list all the varieties they have of said breed. Then when I inquire about them. I discover all of their varieties are running together. There is a huge difference between the offspring being beautifully, colorful. Or the offspring being barnyard mixes!
If by “variety” you mean different color instead of different breed, there are different colors that can be kept together and crosses between colors will simply produce chicks of one parent color or the other. My black and blue breasted red American game bantams are this way. I can cross black and blue breasted red together and get one or the other, with red pyle popping out every so often.