Fall, drop, sink, slump, subside are comparable when they mean to go or to let go downward freely. They are seldom close synonyms, however, because of various specific and essential implications that tend to separate and distinguish them.
Fall, which in the relevant sense is intransitive, suggests a descent by the force of gravity and implies a loss of support opposing gravity; in extended use fall may apply to whatever extends downward or gives an effect of going in a downward direction.
Drop may suggest a falling drop by drop or bit by bit, but usually it stresses a speed, directness, unexpectedness, or casualness in falling or allowing to fall.
Sink fundamentally implies a gradual descending motion, especially into something, often to the point of total submersion, but in frequent somewhat extended use the stress is so strongly on a slow or gradual falling or descent that the notion of submergence is largely or wholly lost.
Slump usually implies a sudden falling or collapsing (as of someone suddenly powerless or suddenly totally enervated).
Subside suggests a gradual descent or return to a normal or usual position, action, or condition after an undue rising, expanding, or boiling up; often it can suggest a sinking below a normal or usual level.