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Phenotypic heterogeneity in cancers is associated with invasive progression and drug resistance. This heterogeneity arises in part from the ability of cancer cells to switch between phenotypic states, but the dynamics of this cellular plasticity remain poorly understood. Here we apply DNA barcodes to quantify and track phenotypic plasticity across hundreds of clones in a population of cancer cells exhibiting epithelial or mesenchymal differentiation phenotypes. We find that the epithelial-to-mesenchymal cell ratio is highly variable across the different clones in cancer cell populations, but remains stable for many generations within the progeny of any single clone—with a heritability of 0.89. To estimate the effects of combination therapies on phenotypically heterogeneous tumours, we generated quantitative simulations incorporating empirical data from our barcoding experiments. These analyses indicated that combination therapies which alternate between epithelial- and mesenchymal-specific treatments eventually select for clones with increased phenotypic plasticity. However, this selection could be minimized by increasing the frequency of alternation between treatments, identifying designs that may minimize selection for increased phenotypic plasticity. These findings establish new insights into phenotypic plasticity in cancer, and suggest design principles for optimizing the effectiveness of combination therapies for phenotypically heterogeneous tumours.