The quantification of the shutdown dose rate (SDR) caused by photons emitted by activated structural materials is an important and necessary step of the design process of fusion energy systems (FES). FES are purposefully designed with modular components that can be moved out of a facility after shutdown for maintenance. It is particularly important to accurately quantify the SDR during maintenance procedures that may cause facility personnel to be in closer proximity to activated equipment. This type of analysis requires neutron and photon transport calculations coupled by activation analysis to determine the SDR. Due to its ability to obtain highly accurate results, the Monte Carlo (MC) method is often used for both transport operations, but the computational expense of obtaining results with low error in systems with heavy shielding can be prohibitive. However, variance reduction (VR) methods can be used to optimize the computational efficiency by artificially increasing the simulation of events that will contribute to the quantity of interest. One hybrid VR technique used to optimize the initial transport step of a multi-step process is known as the Multi-Step Consistent Adjoint Driven Im- portance Sampling (MS-CADIS) method. The basis of MS-CADIS is that the importance function used in each step of the problem must represent the impor- tance of the particles to the final objective function. As the spatial configuration of the materials changes, the probability that they will contribute to the objec- tive function also changes. In the specific case of SDR analysis, the importance function for the neutron transport step must capture the probability of materials to become activated and subsequently emit photons that will make a significant contribution to the SDR. The Groupwise Transmutation (GT)-CADIS method is an implementation of MS-CADIS that optimizes the neutron transport step of SDR calculations. GT-CADIS generates an adjoint neutron source based on certain assumptions and approximations about the transmutation network. This source is used for adjoint transport and the resulting flux is used to generate the biasing parameters to optimize the forward neutron transport. For systems that undergo movement, a new hybrid deterministic/MC VR technique, the Time-integrated (T)GT-CADIS method, that adapts GT-CADIS for dynamic systems by calculating a time-integrated adjoint neutron source was developed. This work demonstrates the tools and workflows necessary to efficiently calculate quantities of interest resulting from coupled, multi-physics processes in dynamic systems.