Energy-efficient computing paradigms beyond conventional von-Neumann architecture, such as neuromorphic computing, require novel devices that enable information storage at nanoscale in an analogue way and in-memory computing. Memristive devices with long-/short-term synaptic plasticity are expected to provide a more capable neuromorphic system compared to traditional Si-based complementary metal-oxide-semiconductor circuits. Here, compositionally graded oxide films of Al-doped MgxZn1−xO (g-Al:MgZnO) are studied to fabricate a memristive device, in which the composition of the film changes continuously through the film thickness. Compositional grading in the films should give rise to asymmetry of Schottky barrier heights at the film-electrode interfaces. The g-Al:MgZnO films are grown by using aerosol-assisted chemical vapor deposition. The current-voltage (I-V) and capacitance-voltage (C-V) characteristics of the films show self-rectifying memristive behaviors which are dependent on maximum applied voltage and repeated application of electrical pulses. Endurance and retention performance tests of the device show stable bipolar resistance switching (BRS) with a short-term memory effect. The short-term memory effects are ascribed to the thermally activated release of the trapped electrons near/at the g-Al:MgZnO film-electrode interface of the device. The volatile resistive switching can be used as a potential selector device in a crossbar memory array and a short-term synapse in neuromorphic computing.