Dafür bin ich nicht zuständig — I am not responsible for that. It is the standard sentence for saying that a matter belongs to someone else.
Es fühlte sich niemand zuständig — nobody felt responsible. Duden gives this as an example sentence for the word.
Das ist nicht mein Bier — that is not my beer. In everyday speech it means “that is not my business.”
Das fällt in mein Ressort — that falls into my department. Duden defines a Ressort as a firmly outlined area of tasks and responsibility.
Kompetenzgerangel and Zuständigkeitsgerangel — wrangling over competence, wrangling over responsibility. Duden marks Kompetenzgerangel as colloquial and disapproving.
Zuständigkeitshalber — for reasons of responsibility. The word belongs to the German of official letters.