We consider the problem of cooperative manipulation by a mobile multi-manipulator system operating in obstacle-cluttered and highly constrained environments under spatio-temporal task specifications. The task requires transporting a grasped object while respecting both continuous robot dynamics and discrete geometric constraints arising from obstacles and narrow passages. To address this hybrid structure, we propose a multi-rate planning and control framework that combines offline generation of an STL-satisfying object trajectory and collision-free base footprints with online constrained inverse kinematics and continuous-time feedback control. The resulting closed-loop system enables coordinated reconfiguration of multiple manipulators while tracking the desired object motion. The approach is evaluated in high-fidelity physics simulations using three Franka Emika Panda mobile manipulators rigidly grasping an object.