**Core Concept**
The key concept here is the anatomical relationship of cervical swellings to the hyoid bone and the pharynx during swallowing. Swallowing causes upward movement of the hyoid bone, which pulls on the thyroid and branchial structures. Only certain swellings move with deglutition due to their attachment or origin.

**Why the Correct Answer is Right**
A branchial cyst arises from the branchial arches and is typically located in the anterior neck, often near the angle of the jaw. It is not attached to the hyoid bone and does not move with deglutition because it lacks direct connection to the pharyngeal structures. In contrast, thyroglossal cysts, thyroid swellings, and tuberculous lymph nodes are either attached to the thyroid or move with the hyoid bone during swallowing. Thus, branchial cysts remain fixed.

**Why Each Wrong Option is Incorrect**
Option A: Thyroglossal cysts move with deglutition because they are linked to the hyoid bone and follow its upward movement during swallowing.
Option B: Thyroid swelling moves with deglutition due to its direct attachment to the hyoid bone via the thyroglossal ligament.
Option D: Tuberculous lymph nodes, especially in the cervical chain, are often located near the thyroid and move with deglutition due to their proximity and mobility.

**Clinical Pearl / High-Yield Fact**
Remember: "Swallowing test" helps differentiate branchial cysts from other cervical swellings. If a swelling does not move with deglutition, it is likely a branchial cyst — a classic exam trap.

✓ Correct Answer: C. Branchial cyst