System dynamic performance needs to consider the influences of the time-varying system performance and various uncertainties in engineering. The evolving process of response affects the safety evaluation of mechanical system with uncertain factors. Considering the parametric randomness, dynamic performance and material strength. a dynamic reliability model is proposed to evaluate the system characteristic. The failure domain is regarded as the dynamic extreme value of system response exceeding the threshold in operation. The dynamic extreme distribution is investigated to measure the probability and parametric sensitivity that a component or system operates securely. According to the saddle-point approximation method, dynamic evolving process is proposed to analyse the cumulative distribution function of system response. According to Poisson process of extreme distribution, the time-continuous dynamic reliability model is established to evaluate the system performance under stochastic load. Additionally, the reliability-based sensitivity model is proposed to investigate each parameter's influence on the system behaviour. The influence degree of each parameter's fluctuation can be quantified based on the indices of dynamic mean-value reliability-based sensitivity and standard deviation reliability-based sensitivity. Finally, an engineering case is taken as an example to analyse the dynamic reliability and reliability-based sensitivity indices. The crude Monte Carlo simulation is taken as the benchmark to validate the reasonability.