A detailed understanding of the resistive switching mechanisms that operate in redox-based resistive random-access memories (ReRAM) is key to controlling these memristive devices and formulating appropriate design rules. Based on distinct fundamental switching mechanisms, two types of ReRAM have emerged: electrochemical metallization memories, in which the mobile species is thought to be metal cations, and valence change memories, in which the mobile species is thought to be oxygen anions (or positively charged oxygen vacancies). Here we show, using scanning tunnelling microscopy and supported by potentiodynamic current-voltage measurements, that in three typical valence change memory materials (TaOx, HfOx and TiOx) the host metal cations are mobile in films of 2 nm thickness. The cations can form metallic filaments and participate in the resistive switching process, illustrating that there is a bridge between the electrochemical metallization mechanism and the valence change mechanism. Reset/Set operations are, we suggest, driven by oxidation (passivation) and reduction reactions. For the Ta/Ta2O5 system, a rutile-type TaO2 film is believed to mediate switching, and we show that devices can be switched from a valence change mode to an electrochemical metallization mode by introducing an intermediate layer of amorphous carbon.