The instability and propagation characteristics of dust-acoustic waves (DAWs) are investigated in a collisionless, magnetized dusty plasma system composed of cold and hot dust grains, along with superthermal ions and electrons. By employing the reductive perturbation technique (RPT), a Zakharov–Kuznetsov (ZK) equation is derived to investigate the growth rate and instability conditions of dust-acoustic (DA) solitary waves. The effects of the superthermal parameter, magnetic field strength, dust charge number, and the densities of hot and cold dust grains on the formation of both compressive and rarefactive solitary structures are examined. Their influence on the growth rate of DA instabilities is also analyzed. The results of our current study will help to clarify the emergence and evolution of nonlinear phenomena in dusty plasma in laboratory experiments and astrophysical environments such as Saturn’s rings, Jupiter’s magnetosphere, and Earth’s mesosphere.