【drop down】 —

1. come down suddenly:

• I heard a noise from the well and saw that the cat had dropped down.

2. bend downwards; sink down:

• When they saw the pursuers they dropped down to take cover.

3. (coll.) pay an unexpected visit:

• Let’s 【drop down】 to his summer home and see if he’s there.

【fall down】 —

1. = 【drop down】 1:

• The hunter fired and the deer fell down dead.

2. (of a building, etc.) collapse:

• The steel scaffolding fell down and narrowly missed a couple of bystanders.

3. (of a project, etc.) be unsuccessful; fail:

• His plan fell down when it proved too costly.