This article presents a novel design framework for topology and component sizing optimization of multi-domain dynamic systems described by conservation laws. Multidisciplinary design optimization (MDO) is a powerful tool for minimizing metrics such as inefficiency and cost for these systems-of-systems (SoS). However, quality of the designs identified from the optimization procedure depends on model accuracy and ability to capture inter-system interactions. This work utilizes a conservation-based, graphical modeling approach to capture physical system dynamics and interactions, and expands it to be used in MDO techniques. This yields three contributions to the literature. First, an augmented graph-based model is provided, expressing continuous and discrete design variable values as changes to vertex size, edge size, and edge connections of the dynamic system graph. Second, a sizing and topology optimization framework is developed using the augmented graph-based model as a basis. Third, analytical and numerical sensitivity functions are derived for a cooling system design problem, stemming from application of the design framework. The design framework is applied to two case studies for cooling subsystem design and electric vehicle (EV) powertrain design, with the goal of optimizing thermal and electrical component sizes, as well as discrete choices in the topology of the system being designed. These case studies provide examples for how the design framework enables analysis of alternatives (AoA) during early design stages.