Usage of 'run something down south'

I have the following sentence which describes a person putting a finger on someone’s chest and trailing it downwards:

• She put a finger lightly on my chest and then ran it down south.

In this context, would it be correct to use run something down south or should it be replaced with trailed it downwards?

• She put a finger lightly on my chest and then trailed it downwards.

If south can mean lower part of the body then I can definitely see the first example working too. I haven’t found any examples of this occurring so probably such definition of the word south does not exist.