The present work studies the double-poles and triple-poles soliton solutions of the Kundu-type equation with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) via the Riemann-Hilbert (RH) method. We construct the RH problem with ZBCs and NZBCs both analyzing the discrete spectral and combining with the analyticity, symmetries, as well as asymptotic behavior of the modified Jost function and the scattering matrix. In the case that the reflection coefficient is double-poles and triple-poles, the inverse scattering transformation (IST) are established and solved by the RH problem with ZBCs and NZBCs, and the reconstruction formula, trace formula and theta conditions. The general formulas of double-poles and triple-poles soliton solutions with ZBCs and NZBCs are explicitly realized through expresses of determinants. The dynamic analysis for the double-poles and triple-poles soliton solutions of ZBCs/NZBCs are vividly described in the form of images.