Nonlinear properties of dust-magnetosonic (DMS) waves are studied in a multispecies thermal dusty plasma, containing electrons, ions, and negatively charged dust particles. In this context, electrons and ions are assumed as inertialess, whereas dust particles are considered mobile or fluid. Solving together the coupled set of fluid-Maxwell equations and making use of the diagonalization matrix technique, the inviscid Burgers's equations are derived and analyzed both analytically and numerically. A solitary solution is used as an initial condition at τ=0, resulting in the formation of localized symmetric pulses, which can develop into small-scale nonstationary (asymmetric) shocklets with temporal progression (τ>0). This leads to the nonlinear self-steepening of waves, propagating in the form of dust-fluid speed and magnetic field profiles in a dusty magnetoplasma. It is found that self-compression of the dust-fluid speed and magnetic field profiles occurs in the direction of propagation and is significantly influenced by the plasma parameters, including the plasma beta, the ion-to-electron temperature ratio, and ion-to-dust density ratio. Additionally, the effective phase and shock speeds are analytically analyzed. The present findings may prove useful for understanding the nonlinear characteristics of DMS waves in a plasma, where dust particulates play a significant role in the self-compression of waves due to the nonlinearity effect.