  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.
