Correct-by-construction manipulation planning in a dynamic environment, where other agents can manipulate objects in the workspace, is a challenging problem. The tight coupling of actions and motions between agents and complexity of mission specifications makes the problem computationally intractable. This paper presents a reactive integrated mission and motion planning for mobile-robot manipulator systems operating in a partially known environment. We introduce a multi-layered synergistic framework that receives high-level mission specifications expressed in linear temporal logic and generates dynamically-feasible and collision-free motion trajectories to achieve it. In the high-level layer, a mission planner constructs a symbolic two-player game between the robots and their environment to synthesis a strategy that adapts to changes in the workspace imposed by other robots. A bilateral synergistic layer is developed to map the designed mission plan to an integrated task and motion planner, constructing a set of robot tasks to move the objects according to the mission strategy. In the low-level planning stage, verifiable motion controllers are designed that can be incrementally composed to guarantee a safe motion planning for each high-level induced task. The proposed framework is illustrated with a multi-robot warehouse example with the mission of moving objects to various locations.