Memristive devices have attracted intensive attention in developing hardware neuromorphic computing systems with high energy efficiency due to their simple structure, low power consumption, and rich switching dynamics resembling biological synapses and neurons in the last decades. Fruitful demonstrations have been achieved in memristive synapses neurons and neural networks in the last few years. Versatile dynamics are involved in the data processing and storage in biological neurons and synapses, which ask for carefully tuning the switching dynamics of the memristive emulators. Note that switching dynamics of the memristive devices are closely related to switching mechanisms. Herein, from the perspective of switching dynamics modulations, the mainstream switching mechanisms including redox reaction with ion migration and electronic effect have been systemically reviewed. The approaches to tune the switching dynamics in the devices with different mechanisms have been described. Finally, some other mechanisms involved in neuromorphic computing are briefly introduced.