  Entanglement breaking channels play a significant role in quantum information
theory. In this work we investigate qubit channels through their property of
`non-locality breaking', defined in a natural way but within the purview of
CHSH nonlocality. This also provides a different perspective on the
relationship between entanglement and nonlocality through the dual picture of
quantum channels instead of through states. For a channel to be entanglement
breaking it is sufficient to `break' the entanglement of maximally entangled
states. We provide examples to show that for CHSH nonlocality breaking such a
property does not hold in general, though for certain channels and for a
restricted class of states for all channels this holds.We also consider
channels whose output remains local under SLOCC and call them `strongly
non-locality breaking'.
  We provide a closed form necessary-sufficient condition for any two-qubit
state to show hidden CHSH nonlocality, which is likely to be useful for other
purposes as well. This in turn allows us to characterize all strongly
non-locality breaking qubit channels. It turns out that unital qubit channels
breaking nonlocality of maximally entangled states are strongly non-locality
breaking while extremal qubit channels cannot be so unless they are
entanglement breaking.
