Inbreeding isn't a mechanism to generate new variation. Inbreeding can exaggerate already existing variation by producing populations with homozygosity for particular alleles, inbreeding is especially important in maintaining a phenotype for which the allele is reccessive. The point is not that dogs in general have been inbred which has lead to an increase in variation, it is that inbreeding of specific populations has lead to a marked decrease in variation in those populations.
Dogs are particularly diverse because artificial selection
can be directed, in the way natural selection is not, and can bring to bear selective pressures much higher than natural selection might normally on very modest changes in phenotype.
Inbreeding isn't the cause of the mutations it simply increases the frequency of the mutation in the population, in the wild this effect would be due to more individuals posessing a beneficial mutation surving and reproducing and a subsequently higher representation of that particular allele in the following generations. Obviously the smaller and more inbred a population the quicker these beneficial alleles are likely to fixate, but also in a small inbred population there are problems with deleterious recessives and genetic drift leading to random fixation of alleles.