Applying the method of moments to the chemical master equation appearing in stochastic chemical kinetics often leads to the so-called closure problem. Recently, several authors showed that this problem can be partially overcome using moment-based semidefinite programs (SDPs). In particular, they showed that moment-based SDPs can be used to calculate rigorous bounds on various descriptions of the stochastic chemical kinetic system’s stationary distribution(s)—for example, mean molecular counts, variances in these counts, and so on. In this paper, we show that these ideas can be extended to the corresponding dynamic problem, calculating time-varying bounds on the same descriptions.