To address the low efficiency and the difficulty of balancing high success rate with motion smoothness in robotic end-to-end dynamic grasping tasks, a robotic end-to-end dynamic grasping method based on Curriculum Reinforcement Learning (CRL) was proposed. First, a multi-modal input network that fuses color images, depth maps, and robot proprioceptive states was constructed to map raw sensory data directly to continuous action commands for the end-effector. Second, a curriculum mechanism with synchronously increasing difficulty and smoothness constraints was designed, and combined with a staged reward function, the agent was guided to master grasping from static to dynamic ones progressively. Finally, Domain Randomization (DR) was employed to enhance the policy's transfer capability of Simulation-to-Reality (Sim-to-Real). Simulation results show that the proposed method achieves a grasping success rate of nearly 100% at target speeds ranging from 0.15 to 0.40 m/s, elevating the upper speed limit for stable grasping from 0.25 m/s of the object detection-based baseline method to 0.40 m/s. Compared to Simple Curriculum Learning (SimpleCL) with only increasing difficulty, the proposed method increases the success rate by 3.6 percentage points and reduces the average joint acceleration and jerk norm by 58.34% and 69.25%, respectively, in the most difficult test. In physical experiments, the grasping success rates of the proposed method for static scene and two dynamic scenes are 95.0%, 90.0%, and 70.0%, respectively. It can be seen that this method effectively coordinates success rate and smoothness in robotic dynamic grasping tasks by jointly optimizing task difficulty and behavioral constraints.