The magnetovolume effects (MVEs) of $\mathrm{M}{\mathrm{n}}_3\mathrm{G}{\mathrm{a}}_{1-x}\mathrm{S}{\mathrm{n}}_x\mathrm{N}$ antiperovskite compounds have been investigated by means of neutron powder diffraction. Increasing the Sn-doping content at the Ga site leads to the broadening of the magnetic phase transition temperature range and the thermal expansion behavior changes from negative to positive. We establish the relationship between the square of the ordered magnetic moment $m^2$ and the volume variation $\mathrm{\Delta }{\omega }_m$ for the antiferromagnetic phase (${\mathrm{\Gamma }}^{5g}$ magnetic structure with rhombohedral symmetry $R\overline{3}m$). The temperature variations of $\mathrm{\Delta }{\omega }_m\left(T\right), m^2\left(T\right)$ and the magnetoelastic coupling constant $C\left(T\right)$ are also quantitatively analyzed according to the itinerant-electron theory. Moreover, the increase of the phonon contribution to the thermal expansion induced by Sn doping and the corresponding decrease of dm/dT are revealed to be the key parameters for tuning the MVEs. Our results allow elucidating and quantifying the mechanism of the spin-lattice coupling and can be used to design magnetic functional materials with controlled thermal expansion behaviors for specific applications.