You probably have Seborrehic Dermatits, and not dry skin.
Densely haired areas are also rich in sebum glands. Seb Derm occurs when there is excess sebum on the skin. Excess sebum provides a good environment for a species of fungi (Malassezia furfur}. These fungi excrete enzymes which break down the sebum in order to isolate the oleic acid which is their food. Free fatty acids are irritating to the skin.
The mechanism of action is unclear, but either the enzymes of the fungi and/or the free fatty acids disrupt the normal turnover rate of the epidermis. Normally a newly formed skin cell at the base of the epidermis takes about 21 days to migrate to the top most layer and slough off in hardly visible/invisible pieces. In this case though, the process is speeded up and the sloughing of cells occurs in larger groups, which are noticed as dandruff.
There is no existing cure for this problem, just ways to manage it. Keeping the skin clear of sebum is difficult under the hair of a beard or mustache. So killing the fungi is the easiest option.
@Nimis has a good suggestion re: dandruff shampoo, but the normal additive, zinc pyrithione, can be very irritating, especially in the mustache zone. You might try “Nizoral”. Very gentle and a bit pricey. In either case, you do need to leave these shampoos on for 5 or so minutes.
For economy, it’s best to prewash these areas with your normal shampoo and then use the medicated shampoo. Using a brush in the prewash to help dislodge particles of skin that are close to becoming free from the skin. Both shampoos work in that they kill the fungi, though Nizoral is more effective and has a longer effect.
You’ll find that, with experimentation, you won’t need to use these “medicated” shampoos every day.
You should wash beard/mustache area with a detergent based cleanser (soap, or liquid cleansers for oily skin 2x/day.