Whole-body reactive obstacle avoidance for mobile manipulators (MM) remains an open research problem. Control Barrier Functions (CBF), combined with Quadratic Programming (QP), have become a popular approach for reactive control with safety guarantees. However, traditional CBF methods often face issues such as pseudo-equilibrium problems (PEP) and are ineffective in handling dynamic obstacles. To overcome these challenges, we introduce the Adaptive Cyclic Inequality (ACI) method. ACI takes into account both the obstacle's velocity and the robot's nominal control to define a directional safety constraint. When added to the CBF-QP, ACI helps avoid PEP and enables reliable collision avoidance in dynamic environments. We validate our approach on a MM that includes a low-dimensional mobile base and a high-dimensional manipulator, demonstrating the generality of the framework. In addition, we integrate a simple yet effective method for avoiding self-collisions, allowing the robot enabling comprehensive whole-body collision-free operation. Extensive benchmark comparisons and experiments demonstrate that our method performs well in unknown and dynamic scenarios, including difficult tasks like avoiding sticks swung by humans and rapidly thrown objects.