This expression is an interesting one, which I’ve seen used in a unique way in both real life and in a Japanese movie.
The original meaning is pretty easy to guess – it’s the term for when passing a baton on in a relay race from one runner to the next. However the meaning has been extended to mean the act of passing off a job or some type of responsibility to an other person.
I saw this expression exchanged (no pun intended) between two Japanese parents, when one parent passes off the responsibility of watching the child to the other parent.
It’s interesting how so many loan words in Japanese have taken their original meaning and went off in a slightly different direction over time. It’s as if the words and phrases themselves are evolving along with the cultures that use them.