The key point here is that "heritable characteristics" are a consequence of the expression of genes. If the genes change, the characteristics will change. In wild animal populations, genes that produce healthy, functional animals are passed to the next generation of offspring. Genes that create deficits of some sort result in offspring that are not as successful and those animals and their genes are eventually eliminated.
For characteristics to change, the genes must change. Likewise, if the genes change, the characteristics must change. Natural selection is this process as it occurs in animal populations, and changes in the characcteristics of those populations over time are the result of changes in the gene pool, the process we call evolution.
Breeding domestic animals is all about managing evolution, in that the purpose is to create individuals that will have the genes that produce the desired characteristics.
This is accomplished in domestic animals the same way it is in nature, by preferentially breeding the animals with the favored characteristrics and therefore, the favored genes. Once the animals have the preferred traits, the process changes from guided change to protecting the status quo, i.e., freezing those traits in the animals by preventing changes in their genes. If you can prevent the genes in the animals from changing over the generations, the phenotype - both what you can see and what you can't - should also stay the same.
The problem should be obvious. Say you have a population of healthy dogs with great type and health, and you want them to stay that way in subsequent generations. How could you do this? Well, keeping the genes for type isn't so hard. Remember, most of the critical genes for type are fixed (i.e., homozygous and the same in every dog) early in the development of a breed, and you can maintain them by selection based on physical inspection. But for all the other "invisible" traits (e.g., physiology), you need to be selecting for "invisible" genes, because as we have seen, to have the characteristics, you must also have the genes. Right. We get that. But again, how do you do this?
Everybody should be able to see the problem. To breed sustainably, you need to start with dogs that have the genes for your perfect example of the breed - both inside and out - and breed in a way that prevents the gene pool from changing over time. For this, all the genes in the dogs of the present generation must be replicated and packaged into the offspring produced for the next generation. The particular genes each individual inherits will be different, but the frequencies of the genes in the gene pool should stay the same over time. For all those things that are a consequence of many genes, some of which might interact, getting this transfer of the gene pool into the next generation is critical. If the next generation includes only part of the gene pool, then you won't get the same traits. Remember, for same traits, you need the same genes. If the genes change, the traits will change.
Breeders of commercial domestic animals know this. They develop breeding strategies to prevent their gene pool from changing over time. This is the purpose of rotational breeding. If you want to breed sustainably, you must take steps to prevent the gene pool from changing over time.
The dog fancy does not do this. We select for the traits we can assess, with no worry about protecting the genes for the traits we can't. In short, we do not breed in a way that will prevent the gene pool from changing over time. In fact, we apply strong selection for those evident traits by breeding the individuals we deem the "best", and remove the other animals from the gene pool. Consequently, the composition of the gene pool is different in every generation, even if the dogs look physically the same. That is what we are referring to when we talk about loss of genetic diversity. Furthermore, not only the composition of the gene pool changes, but the expression of those genes changes because breeding to related animals produces inbreeding, which is an increase in homozygosity. There are always mutations hanging out harmlessly in the genome of every animal, so producing homozygosity is going to result in changes in characteristics. And indeed, this is the endless battle of the breeder, who is breeding only the "best" animals, yet producing animals that are flawed in some way. We understand why this happens.
To keep the dogs the same, both for things we can see and things we can't like health, we need to prevent changes in the genes.
In the dog fancy, this is the elephant in the room. We. Do. Not. Do. This.
The gene pool of every dog breed is changing every generation because we breed only a small fraction of the animals produced by the previous generation. Because genes cannot be added to the gene pool (this is the closed stud book), the gene pool will lose genes that are not replaced. If we believe in genetics (and we do!), every gene has some job it's there to do, and if we remove that gene, we should expect that something will be broken. Sometimes it's something obvious, but mostly these broken things create tiny little problems that escape immediate attention but accumulate over time until we have a real issue. This is not rocket science. A grade school child should be able to understand that if you remove all the red M&Ms from the bowl, you will not be able to eat any red M&Ms.
Okay, so dog breeders have not been protecting the gene pools of their breeds, and we have the problems to show for it. (A point of history: this is a legacy of a culture at the time of breed formation that prioritized "purity" over preservation of gene pools, although to be fair this was long before we understood the genetic basis of inheritance.) If the loss of genes every generation results in changes in function or health, what sense does it make to do more of what caused the problem in the first place? We remove dogs from breeding that don't have the traits or function that we want. We think we're "getting rid of a problem." But the problem is the loss of the diversity of genes that are critical to the function of complex physiology, behavior, and biochemistry. We created our problems by not protecting the gene pool of a breed, and we are trying to return the breed to health by changing the gene pool even more, and in ways that we have no way to know.
Ask our grade school child how we fix this. If the red M&Ms somehow made all the other M&Ms taste better in some invisible way, and the remaining M&Ms just aren't as good without the red ones in the mix, the only way we will get that great taste back is by replacing the red M&Ms that were lost. Now, they will come from a different bag than the original ones. But they're exactly the same. Put those in the bowl and we can recover what we started with. If we for some reason insist that we cannot add any M&Ms from a different bag, then we are permanently stuck with our red M&M deficit and a bowl of inferior-tasting M&Ms.
The health problems in purebred dogs are a consequence of inadequate genetic management that resulted from adopting breeding strategies that do not protect the gene pool. Our inability to solve these problem despite decades of diligent effort is a predictable consequence of breeding strategies that do not restore the gene pool to its original condition. Notice that DNA testing is not going make dogs healthier, because we are still not doing anything to protect the gene pool from deleterious change, much less restoring it to that of a population of healthy dogs.
We must understand genetics to breed dogs. But we must also understand evolution. We do pretty good with the genetics stuff. But are failing miserably at the evolution part, which is the implementation of genetic management. The tool for this is population genetics, something most breeders know nothing about, or understand "just enough to be dangerous," as the saying goes. If we had relatively healthy populations of dogs, a general understanding of population genetics would be fine. But what we have are breeds that have been under strong genetic selection but without genetic management, so gene pools have changed every generation without guidance and with ineffective protection of diversity. These gene pools are well and truly broken.
We will not make dogs healthier by "health testing," and you should now understand why. We are also not doing "preservation breeding," and that you should also understand. Research will not solve the health problems of dogs, because the problem is not the disease, it's the loss of the genes necessary for all the complicated stuff that needs to happen in a dog over an entire lifetime - birth, growth, immune defense, behavior, and an infinity of other events and processes that are the essence of life. If 20% of the genome has been lost from those original dogs that had both good type and good health, we will not have those original traits, and we don't. We have carefully bred to protect the genes for type; we have deliberately bred in a way guaranteed to lose genes for everything else.