Safe reinforcement learning-based robot motion planning demonstrates significant potential in real-world physical environments. However, existing methods exhibit limited stability and adaptability in complex, risk-sensitive scenarios, often failing to meet safety requirements. To address this, we propose a safe reinforcement learning method with temporally dynamic evolution. First, we develop a policy network within a continuous-time ordinary differential equation framework, integrating Euler discretization to transform the policy from a static mapping into a differentiable dynamical system. This approach enhances the modeling of non-uniform temporal sequences, thereby improving adaptability in complex environments. Second, we introduce an adaptive constrained proximal policy optimization algorithm to solve safety-constrained optimization, enabling flexible and safe motion control. Finally, we evaluate the method on the Safety-Gymnasium benchmark and in custom Gazebo physical environments. Experimental results show that the proposed dynamically evolving safe reinforcement learning motion planning method outperforms existing methods, with a 53.7% higher success rate than the baseline PPO-Lag. It achieves the highest average return and success rate across environments of varying difficulty, demonstrating superior planning performance in generalized scenarios.