Circulating tumor cells (CTCs) traverse vessels to travel from the primary tumor to distant organs where they adhere, transmigrate, and seed metastases. To cope with these challenges, CTCs have reached maximal flexibility to change their differentiation status, morphology, migratory capacity, and their responses to genotoxic stress caused by metabolic changes, hormones, the inflammatory environment, or cytostatic treatment. A significant percentage of breast cancer cells are defective in homologous recombination repair and other mechanisms that protect the integrity of the replication fork. To prevent cell death caused by broken forks, alternative, mutagenic repair, and bypass pathways are engaged but these increase genomic instability. CTCs, arising from such breast tumors, are endowed with an even larger toolbox of escape mechanisms that can be switched on and off at different stages during their journey according to the stress stimulus. Accumulating evidence suggests that DNA damage responses, DNA repair, and replication are integral parts of a regulatory network orchestrating the plasticity of stemness features and transitions between epithelial and mesenchymal states in CTCs. This review summarizes the published information on these regulatory circuits of relevance for the design of biomarkers reflecting CTC functions in real-time to monitor therapeutic responses and detect evolving chemoresistance mechanisms.