For a better understanding of multistimuli-responsive caloric materials with a first-order transition and for optimization of their functional properties, it is necessary to predict the behavior of the material under changes of both magnetic field and pressure. Here, we design and build a special device that can provide a self-consistent set of parameters needed for the comprehensive characterization of multistimuli-responsive functional magnetic materials. Using this scientific instrument, a data set of simultaneously measured magnetization, ${M(T)}_{H}$, and volume magnetostriction, $\ensuremath{\omega}(T{)}_{H}$, values are obtained for ${\mathrm{LaFe}}_{11.4}{\mathrm{Si}}_{1.6}$ with a first-order transition. Furthermore, based on simultaneously measured M(T) and \ensuremath{\omega}(T) dependencies obtained at ambient pressure, we develop an approach that allows the behavior of magnetization under different pressures, ${M(T)}_{P}$, to be described analytically. Additional parameters, such as compressibility, \ensuremath{\kappa}(T); thermal expansion coefficient, \ensuremath{\alpha}(T); and magnetoelastic interaction or effective magnetovolume coupling constant, ${C}_{MV}$, are determined. For verification of our developed model, direct measurements of magnetization under external pressure (up to P = 1 GPa) are carried out on the same sample as that used for simultaneous measurement of magnetization and magnetovolume effect. A comparison of simulated ${M(T)}_{P}$ dependencies with experimental $M(T{)}_{P}$ confirms that our approach provides a more realistic behavior of transition temperature under pressure, ${T}_{C}(P)$, than that of the ${T}_{C}(P)$ predicted by the Bean-Rodbell model; thus, this approach is more suitable for predicting the behavior of multistimuli-responsive caloric materials with first-order transitions under changes of both magnetic field and pressure.