This paper proposes a novel magnetic wheeled wall-climbing robot with a passive-compliant suspension mechanism and investigates the variable-curvature adaptability and kinematics transformation flow. The paper reveals the effect of the two wheel–wall gaps on the adsorption force of the wall-climbing robot. Further, a new way of passively adapting to variable-curvature pipe surfaces is presented with a passive-compliant suspension mechanism to ensure all four wheels could be attached simultaneously to the pipe’s inner inclined surfaces simultaneously. The paper further introduces the absorption force evaluation index, which makes the motion characteristics of the robot on the variable-curvatured inclined surfaces quantitatively analyzed, and verifies the variable-curvature adaptability of the robot. Meanwhile, given the dissimilar elliptical configurations with variable curvatures across the cylindrical pipe surfaces, the kinematics transformation flow of the robot is established by combining the motion posture of the robot on the pipe’s inner surface. Finally, the experimental results are given to illustrate the flexible, stable, and reliable motion on the pipe inner surface of the magnetic wheeled wall-climbing robot and to demonstrate the passive-compliant suspension-added robot has a good variable-curvature adaptability.