We study the performance of LDPC codes over the cascaded BSC-BAWGN channel. This channel belongs to a family of binary-input, memoryless, symmetric-output channels, one that we call the {CBMSC(p, σ)} family. We analyze the belief propagation (BP) decoder over this channel by characterizing the decodable region of an ensemble of LDPC codes. We then give inner and outer bounds for this decodable region based on existing universal bounds on the performance of a BP decoder. We numerically evaluate the decodable region using density evolution. We also propose other message-passing schemes of interest and give their decodable regions. The performance of each proposed decoder over the CBMS channel family is evaluated through simulations. Finally, we explore capacity-approaching LDPC code ensembles for the {CBMSC(p, σ)} family.