This paper presents a novel sampling-based motion planning method using bidirectional search with a time-configuration space representation that is able to efficiently generate collision-free trajectories in complex and non-static environments. Our approach exploits time indexing to separate a complex problem with mixed constraints into multiple subproblems with simpler constraints that can be solved efficiently. We further introduce a planning framework by incorporating the proposed planning method enabling efficient pick-and-place of large objects in various scenarios. Simulation as well as hardware experiments show that the method also scales from redundant robot arms to mobile manipulators and humanoids. In particular, we have demonstrated that the proposed method is able to plan collision-free motion for a humanoid robot to pick up a large object placed inside a moving storage box while walking.