Development of drug resistance in cancer has major implications for patients' outcome. It is related to processes involved in the decrease of drug efficacy, which are strongly influenced by intratumor heterogeneity and changes in the microenvironment. Heterogeneity arises, to a large extent, from genetic mutations analogously to Darwinian evolution, when selection of tumor cells results from the adaptation to the microenvironment, but could also emerge as a consequence of epigenetic mutations driven by stochastic events. An important exogenous source of alterations is the action of chemotherapeutic agents, which not only affects the signalling pathways but also the interactions among cells. In this work we provide experimental evidence from in vitro assays and put forward a mathematical kinetic transport model to describe the dynamics displayed by a system of non-small-cell lung carcinoma cells (NCI-H460) which, depending on the effect of a chemotherapeutic agent (doxorubicin), exhibits a complex interplay between Darwinian selection, Lamarckian induction and the nonlocal transfer of extracellular microvesicles. The role played by all of these processes to multidrug resistance in cancer is elucidated and quantified.