A family of higher-order rational lumps on non-zero constant background of Davey–Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of which is markedly distinct from the necessary orthogonal scattering for the higher-order lumps on zero background. Additionally, it is illustrated that the higher-order lumps containing multi-peaked n-lumps can be regarded as a nonlinear superposition of n first-order ones as .