The advent of caloric materials for magnetocaloric, electrocaloric, and elastocaloric cooling is changing the landscape of solid state cooling technologies with potentials for high-efficiency and environmentally friendly residential and commercial cooling and heat-pumping applications. Given that caloric materials are ferroic materials that undergo first (or second) order phase transitions near room temperature, they open up intriguing possibilities for multiferroic devices with hitherto unexplored functionalities coupling their thermal properties with different fields (magnetic, electric, and stress) through composite configurations. Here we demonstrate a magneto-elastocaloric effect with ultra-low magnetic field (0.16 T) in a compact geometry to generate a cooling temperature change as large as 4 K using a magnetostriction/superelastic alloy composite. Such composite systems can be used to circumvent shortcomings of existing technologies such as the need for high-stress actuation mechanism for elastocaloric materials and the high magnetic field requirement of magnetocaloric materials, while enabling new applications such as compact remote cooling devices.