Moment bounding schemes for stationary as well as exit time and location distributions have been successfully employed in a range of applications including analysis of stochastic reaction networks [3, 8, 10, 5, 2], diffusion processes [11, 7] and the pricing of financial derivatives [6, 9]. In contrast, bounding schemes for moment trajectories are less well-established. In this work, we extend the recently proposed semidefinite programming based bounding scheme for transient moment solutions of the Chemical Master Equation by Dowdy and Barton [4]. The main contribution is a novel hierarchy of linear and semidefinite moment conditions which directly reflects temporal causality of the moment trajectories. These constraints allow to tighten the generated bounds considerably while possessing a favorable scaling behavior (linear vs. combinatorial) when compared to the set of moment conditions employed by Dowdy and Barton [4]. Additionally, we present a new bounding scheme based on a polynomial restriction of a related continuous-time optimal control problem. The resultant infinite dimensional optimization problem is reduced to a finite semidefinite program via a sum-of-squares characterizations of matrix polynomials [1] to enforce semidefinite constraints over the continuum of time. As an aside, we note that all results naturally extend to bounding the transient moments associated with stochastic processes described by Itô differential equations with polynomial drift and diffusion coefficients. We illustrate the results with examples drawn from stochastic chemical kinetics and other fields.