  Incompatibility and nonlocality are not only of foundational interest but
also act as important resources for quantum information theory. In the
Clauser-Horne-Shimony-Holt (CHSH) scenario, the incompatibility of a pair of
observables is known to be equivalent to Bell nonlocality. Here, we investigate
these notions in the context of qubit channels. The Bell-CHSH inequality has a
greater perspective--compared to any genuine tripartite nonlocality
scenario--while determining the interplay between nonlocality breaking qubit
channels and incompatibility breaking qubit channels. In the Bell-CHSH
scenario, we prove that if the conjugate of a channel is incompatibility
breaking, then the channel is itself nonlocality breaking and vice versa.
However, this equivalence is not straightforwardly generalized to multipartite
systems, due to the absence of an equivalence relation between incompatibility
and nonlocality in the multipartite scenario. We investigate this relation in
the tripartite scenario by considering some well-known states like
Greenberger-Horne-Zeilinger and $W$ states and using the notion of Mermin and
Svetlichny nonlocality. By subjecting the parties in question to unital qubit
channels, we identify the range of state and channel parameters for which
incompatibility coexists with nonlocality. Further, we identify the set of
unital qubit channels that is Mermin or Svetlichny nonlocality breaking
irrespective of the input state.
