Here is a clear, well-sourced answer:

Yes, but infrequently. Naproxen is an NSAID (non-steroidal anti-inflammatory drug) used for pain and inflammation. According to Goodman & Gilman's Pharmacological Basis of Therapeutics, drowsiness is listed as a CNS side effect occurring in 3%-9% of users - roughly the same rate as with other NSAIDs like indomethacin. So while it is not considered a sedating drug, a minority of people do experience drowsiness when taking it. Other CNS effects include headache (also 3%-9%).

Key point: Naproxen's drowsiness is a side effect, not its intended action. It does not work on the central nervous system to cause sleep.

Generally no. Domperidone is a dopamine antagonist used for nausea, vomiting, and to speed up gastric emptying. The important pharmacological reason it rarely causes sedation is that it does not cross the blood-brain barrier well - it acts mainly on peripheral dopamine receptors in the gut.

For comparison, the textbook Washington Manual of Medical Therapeutics notes that metoclopramide - a very similar drug in the same class - does cause drowsiness and other CNS effects (tardive dyskinesia, parkinsonism). Domperidone was specifically developed as an alternative that avoids those CNS side effects precisely because of its poor penetration into the brain.

Bottom line: If you feel sleepy after taking either of these drugs, naproxen is the more likely culprit. Domperidone is considered one of the safest anti-nausea agents precisely because it lacks the sedating central effects seen with similar drugs like metoclopramide.