As a countermeasure to visual-manual distractions, auditory-verbal (voice) interfaces are becoming increasingly popular for in-vehicle systems. This opens up new opportunities for drivers to receive proactive personalized services from various service domains. However, prior studies warned that such interactions can cause cognitive distractions due to the nature of concurrent multitasking with a limited amount of cognitive resources. In this study, we examined (1) how the varying demands of proactive voice tasks under diverse driving situations impact driver interruptibility, and (2) how drivers adapt their concurrent multitasking of driving and proactive voice tasks, and how the adaptive behaviors are related to driver interruptibility. Our quantitative and qualitative analyses showed that in addition to the driving-task demand, the voice-task demand and adaptive behaviors are also significantly related to driver interruptibility. Additionally, we discuss how our findings can be used to design and realize three types of flow-control mechanisms for voice interactions that can improve driver interruptibility.