Green’s Dictionary of Slang

run down v.

1. [mid-18C+] to denigrate someone, to slander someone.

2. [1940s] (W.I.) to seduce someone, to persuade them to become one’s lover.

3. [1940s+] to rehearse, to practise, to explain.

4. [1960s] to verbalize.

In phrases

run down the game (v.)

see under game n.

run it down (v.) (also run it to)

[1950s+] (mainly US black) to explain, to point out facts.