Single unit follicle transplants can be done as you suggest. Care is taken to match the donor characteristics to the desired recipient site. Pubic hair can be used to fill in thin spots in a beard just as nasal hair is a good match for eyebrows or eyelashes. However, directional growth must match the desired recipient growth to create the most natural effects. Expect that not all of the follicles will survive and that staged procedures may be needed to establish desired density.
Pubic hair transplant to your scalp is possible. However scalp hair is usually the primary choice for donor hair.
Body hair can theoretically be transplanted to another area of the body and it will grow as long as it has an adequate blood supply and is handled appropriately. The two main things to consider are the survival rate (which tends to be lower for body hair) and matching the type of hair (caliber, curl, etc.) to the recipient area. Incidentally, the opposite can be true. Beard hair (and scalp hair) can be used to graft a pubic area (particularly useful in reconstructive cases).Good luck!
Most patients have lower abdominal and pubic hairs that are .03-.05 mm. Beard hair is generally thicker than that. However you do find more double and triple follicular units int he pubic region and those can be separated to provide a higher number of hair count as beard hairs are generally single hair grafts.
I have done a few cases where I used pubic hair as a donor hair in cases of donor area depletion. I don't see a reason not to use it on request for beard provided you are not very young and are yet to grow out your beard.