Hybrid fluid-kinetic models are adopted to investigate the collisionless plasma sheaths with κ-electron distribution and containing dust particles. A modified ensemble-in-spacetime model is utilized. The model is suitable for sheaths with non-Maxwellian electrons, such as, in DIII-D tokamak. In the intermediate radio-frequency regime, the flux and energy of ions are time-modulated within the sheath. Ions are not inertial-less. The value of κ affects the electron and ion density. Time-averaged electron and ion densities, the sheath edge position, and the area of the sheath voltage-sheath charge hysteresis loop increase by increasing the value of κ.The ion energy distribution is not sensitive to the κ distribution. Dust particles are charged with different negative charges based on their radius, their position within the sheath, the κ-electron distribution, and the sheath potential. Dust particles exhibit an oscillatory motion due to the interaction with the plasma and the gravitational fields.