Quadruped robots equipped with manipulators need fast and precise grasping and detection algorithms for the transportation of disaster relief supplies. To address this, we developed a framework for these robots, comprising a Grasp Detection Controller (GDC), a Joint Trajectory Planner (JTP), a Leg Joint Controller (LJC), and a Manipulator Joint Controller (MJC). In the GDC, we proposed a lightweight grasp detection CNN based on DenseBlock called DES-LGCNN, which reduced algorithm complexity while maintaining accuracy by incorporating UP and DOWN modules with DenseBlock. For JTP, we optimized the model based on quadruped robot kinematics to enhance wrist camera visibility in dynamic environments. We integrated the network and model into our homemade robot control system and verified our framework through multiple experiments. First, we evaluated the accuracy of the grasp detection algorithm using the Cornell and Jacquard datasets. On the Jacquard dataset, we achieved a detection accuracy of 92.49% for grasp points within 6 ms. Second, we verified its visibility through simulation. Finally, we conducted dynamic scene experiments which consisted of a dynamic target scenario (DTS), a dynamic base scenario (DBS), and a dynamic target and base scenario (DTBS) using an SDU-150 physical robot. In all three scenarios, the object was successfully grasped. The results demonstrate the effectiveness of our framework in managing dynamic environments throughout task execution.