Mutations are changes to existing functions. To understand these changes you need to first understand the original function.

Fur color is based on melanin. Two types of melanin are produced; black/brown eumelanin and yellow/red pheomelanin. 

Melanin is produced in cells called melanocytes. Inside these cells there are tiny organs (organelles) called melanosomes that build the melanin. 

The melanocytes need to be protected, and remain at the very bottom layer of skin. In order to send their melanin into the next layer, they have branches called dendrites that protrude out of the cell like a tree. Melanocytes send their melanosome organelles into these dendrites, and then up into the cells above them: keratinocytes. 

These keratinocytes form the bulk of the skin. The keratinocytes are like the bricks of a wall, and the melanocytes produce protective paint for them.

The genes listed below are only a few of many that aid in the function of melanin production at different steps in the path.

ASIP Gene

"The viable yellow agouti (Avy) mouse model, in which coat color variation is correlated to epigenetic marks established early in development, has been used to investigate the impacts of nutritional and environmental influences on the fetal epigenome (Fig. 1A and B)."
Source

DNA methylation (markers on top of the DNA, epigenetics) is responsible for brindle stripes. Lots of methylation = overmarked, little methylation = undermarked.

Epigenetics is altered by the environment: diet, stress, temperatures, etc. Since the fetuses share an environment with the dam during development, her epigenetics will be more likely what the pups express. That's why brindle usually takes after the dam more so than the sire.

"The Avy allele is the most extensively studied murine metastable epiallele. Metastable epialleles are identical alleles that are variably expressed due to epigenetic modifications that are established very early in development."
Source

However, the mouse agouti locus methylation patterns have also proven to be directly inheritable.

"Epigenetic modifications have effects on phenotype, but they are generally considered to be cleared on passage through the germ line in mammals, so that only genetic traits are inherited. Here we describe the inheritance of an epigenetic modification at the agouti locus in mice.
...
We demonstrate here that this maternal epigenetic effect is not the result of a maternally contributed environment. Rather, our data show that it results from incomplete erasure of an epigenetic modification when a silenced Avy allele is passed through the female germ line, with consequent inheritance of the epigenetic modification."
Source

The D locus gene Myo5a is responsible for producing tiny transport proteins that help move the melanin containing melanosomes into the dendrites of the melanocyte cell. (See Important Info above.)

The blue dilute mutation causes fewer transport proteins to function properly. This means less melanin goes where it's supposed to. Overall pigment is lessened, both black and yellow. 

While the mouse embryo is developing, not all of the melanoblasts finish migrating to all parts of the body. Melanoblasts later turn into melanocytes (melanin producing cells). The feet and tail tips tend to be the last place they migrate to causing these areas to be void of pigment entirely and resulting in white fur and skin. (1) The genes associated with marked varieties are the same genes responsible for various steps of this melanoblast migration or conversion into viable melanocytes. (2)

(1) Sheila Schmutz, Ph.D.
(2) Coat Color Mutations, Animals