In this paper, we investigate a (3+1)-dimensional generalized variable-coefficient shallow water wave equation, which can be used to describe the flow below a pressure surface in oceanography and atmospheric science. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the semi-rational solutions which describe the lumps and rogue waves interacting with the kink solitons. We find that the lump appears from one kink soliton and fuses into the other on the x?y and x?t planes. However, on the x?z plane, the localized waves in the middle of the parallel kink solitons are in two forms: lumps and line rogue waves. The effects of the variable coefficients on the two forms are discussed. The dispersion coefficient influences the speed of solitons, while the background coefficient influences the background’s height.