Integrated terrestrial and non-terrestrial networks (ITNTNs) are regarded as a key architectural paradigm for sixth-generation (6G) wireless systems. This paper investigates a dual-aerial reconfigurable intelligent surface (RIS)-assisted ITNTN, where a terrestrial base station (TBS) and a satellite (SAT) jointly serve terrestrial and satellite users with the aid of an unmanned aerial vehicle (UAV)-mounted RIS and a high-altitude platform (HAP)-mounted RIS. We formulate an average sum-rate maximization problem by jointly optimizing the TBS and SAT precoders, the RIS phase shift matrices, and the three-dimensional trajectories of the UAV and the HAP, subject to transmit power, unit-modulus, and mobility constraints. The resulting optimization problem is highly non-convex due to the strong coupling among the transmit precoders, RIS phase shifts, and aerial platform mobility. To efficiently address this challenge, we propose a block coordinate descent (BCD) framework that integrates weighted minimum mean square error (WMMSE) optimization for precoder design, a manifold-based Riemannian conjugate gradient (RCG) method for RIS phase-shift optimization, and successive convex approximation (SCA) for trajectory optimization. The proposed algorithm is shown to converge to a stationary point. The simulation results show that the proposed joint design achieves an approximately $7.05 \%$ higher average sum-rate compared to the random RIS scheme, highlighting the effectiveness of dual-aerial RIS deployment and joint communication-mobility optimization in ITNTNs.