**Core Concept**
Special visceral efferent (SVE) fibers are a type of motor nerve fiber involved in the innervation of muscles derived from the branchial arches. These fibers play a crucial role in the control of muscles responsible for facial expressions, mastication, and other functions related to the branchial arches.

**Why the Correct Answer is Right**
The dorsal nucleus of the X cranial nerve, also known as the vagus nerve, is primarily involved in the transmission of general visceral efferent (GVE) fibers, which are responsible for the innervation of smooth muscle and glands. The GVE fibers of the vagus nerve regulate various autonomic functions, including heart rate, digestion, and respiration. In contrast, SVE fibers are involved in the innervation of skeletal muscles derived from the branchial arches.

**Why Each Wrong Option is Incorrect**
**Option A:** The nucleus ambiguus is indeed involved in the transmission of SVE fibers, which innervate muscles derived from the branchial arches, including those responsible for swallowing and vocal cord movement.

**Option B:** The motor nucleus of the V cranial nerve, also known as the trigeminal nerve, is involved in the transmission of SVE fibers that innervate the muscles of mastication, including the temporalis, masseter, and medial pterygoid muscles.

**Option C:** The motor nucleus of the VII cranial nerve, also known as the facial nerve, is involved in the transmission of SVE fibers that innervate the muscles of facial expression, including the orbicularis oculi, zygomaticus major, and platysma muscles.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that SVE fibers are involved in the innervation of muscles derived from the branchial arches, while GVE fibers are involved in the innervation of smooth muscle and glands.

**✓ Correct Answer: D. Dorsal nucleus of X cranial nerve**