***Neither a teacher nor a native speaker.***
I believe that "the kill" and "the visit" refers to one.
That means you're talking about one (specific) kill or visit.
"The killing" and "the visiting" means there are several ones.
Not every visit is free.
However, the visit tomorrow will be free.
The visitting is free.
This means it is always free.
P.S. Keep in mind that it was only a guess, and maybe I'm totally wrong here...
- For Teachers