Coupled order parameters in phase-transition materials can be controlled using various driving forces such as temperature, magnetic and electric field, strain, spin-polarized currents and optical pulses. Tuning the material properties to achieve efficient transitions would enable fast and low-power electronic devices. Here we show that the first-order metamagnetic phase transition in FeRh films becomes strongly asymmetric in mesoscale structures. In patterned FeRh stripes we observed pronounced supercooling and an avalanche-like abrupt transition from the ferromagnetic to the antiferromagnetic phase, while the reverse transition remains nearly continuous over a broad temperature range. Although modest asymmetry signatures have been found in FeRh films, the effect is dramatically enhanced at the mesoscale. The activation volume of the antiferromagnetic phase is more than two orders of magnitude larger than typical magnetic heterogeneities observed in films. The collective behaviour upon cooling results from the role of long-range ferromagnetic exchange correlations that become important at the mesoscale and should be a general property of first-order metamagnetic phase transitions.