This paper presents a multi-contact motion adaptation framework that enables teleoperation of high degree-of-freedom (DoF) robots, such as quadrupeds and humanoids, for loco-manipulation tasks in multi-contact settings. Our proposed algorithms optimize whole-body configurations and formulate the retargeting of multi-contact motions as sequential quadratic programming, which is robust and stable near the edges of feasibility constraints. Our framework allows real-time operation of the robot and reduces cognitive load for the operator because infeasible commands are automatically adapted into physically stable and viable motions on the robot. The results in simulations with full dynamics demonstrated the effectiveness of teleoperating different legged robots interactively and generating rich multi-contact movements. We evaluated the computational efficiency of the proposed algorithms, and further validated and analyzed multi-contact loco-manipulation tasks on humanoid and quadruped robots by reaching, active pushing and various traversal on uneven terrains.