We report appearance of the net magnetization in Mn-based antiperovskite compounds as a result of the external strain gradient (flexomagnetic effect). In particular, we describe the mechanism of the magnetization induction in the ${\text{Mn}}_{3}\text{GaN}$ at the atomic level in terms of the behavior of the local magnetic moments of the Mn atoms. We show that the flexomagnetic effect is linear and results from the nonuniformity of the strain, i.e., it is absent not only in the ground state but also when the applied external strain is uniform. We estimate the flexomagnetic coefficient to be $\ensuremath{\sim}2\text{ }{\ensuremath{\mu}}_{B}\text{ }\text{\AA{}}$. We show that at the moderate values of the strain gradient $(\ensuremath{\sim}0.1\mathrm{%})$ the flexomagnetic contribution to the net induced magnetization is comparable with the nonlinear contribution. Finally, we apply a classical Heisenberg model to study the correlation between spin-exchange interaction and flexomagnetism, using time quantified Monte Carlo simulation. This confirms the linear nature of the flexomagnetic effect and helps understanding phenomenological aspects behind it.