(Disclaimer: I'm not a breeder, so I can only write about my observations.)
it seems like shopping for sires from other reputable breeders, rather than continuing to line breed one's own dogs, on a regular basis would thin each breeder's gene pool in a good way.
Most reputable basset breeders, with the exception of a handful of larger kennels, keep only one stud dog at a time, but they generally keep and breed several bitches during a given interval. They almost never breed all their bitches to their single stud dog; this type of practice, in my experience, is more typical of a BYB or commercial breeder. Instead, their stud dog is used by other breeders, and they use other breeders' stud dogs, as you mentioned. These breeders may have lines that are related to a greater or lesser degree, but not always. In my dozen or so years in the breed, some of the biggest winners have been outcrosses.
If you can stomach it

, here's another link, an
Introduction to Population Genetics. Although it talks about the importance of heterozygosity (diversity) it also includes the following section.
Out-breeding Depression
In some cases, inbreeding is purposely done to create a \"pure line\" of individuals which all have similar genes. This is the case with many domesticated species such as pure-bred pets, laboratory animals, and agricultural plants and animals. Breeders intentionally inbreed to create a population that has the genes for desired traits in a homozygous state, so that the offspring will perpetuate the desired phenotype. Over generations, individuals with deleterious genes are removed from the breeding population, resulting in a monomorphic, yet healthy population. These populations can actually suffer \"out-breeding depression\" when crossed to unrelated individuals which results in new combinations of alleles, and possible entry of deleterious genes.
Inbreeding or \"line-breeding\" as it is referred to by pet breeders, must be done very carefully to prevent the creation of a line that is almost perfect but contains a major flaw, such as domestic dog breeds that have the propensity for hearing problems (Dalmatians, cocker spaniels) or seizures (Irish setters). Responsible breeders will often discontinue breeding a line whenever the slightest chance of a deleterious gene is noticed, sacrificing many years of work to prevent damaging the breed.
When I said this was the topic of debate, I was guilty of understatement. I've read several flame wars between proponents of line-breeding and proponents of assortive mating, and if I were to start breeding I'd probably incorporate both practices into my program, depending on my goals for a specific breeding.
However, this may actually be something of a moot discussion...a preliminary investigation finds that numbers of MHC alleles, while quite numerous and varied between breeds, are relatively limited within breeds. We may be trying to close the barn door after the horse has escaped.
Kennedy LJ, Barnes A, Happ GM, Quinnell RJ, Bennett D, Angles JM, Day MJ, Carmichael N, Innes JF, Isherwood D, Carter SD, Thomson W, Ollier WE.
Extensive interbreed, but minimal intrabreed, variation of DLA class II alleles and haplotypes in dogs.Tissue Antigens. 2002 Mar;59(3):194-204.
An interesting speculation in the above abstract:
The high interbreed, and relatively low intrabreed, variation of MHC alleles and haplotypes found in this study could provide an explanation for reports of interbreed variation of immune responses to vaccines, viruses and other infections.