We have studied the barocaloric effect (BCE) in the geometrically frustrated antiferromagnet Mn$_{3}$NiN across the N{e}el transition temperature. Experimentally we find a larger barocaloric entropy change by a factor of 1.6 than that recently discovered in the isostructural antiperovskite Mn$_{3}$GaN despite greater magnetovolume coupling in the latter. By fitting experimental data to theory we show that the larger BCE of Mn$_{3}$NiN originates from multi-site exchange interactions amongst the local Mn magnetic moments and their coupling with itinerant electron spins. Using this framework, we discuss the route to maximise the BCE in the wider Mn$_{3}$AN family.
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