When adopting a deep learning model for embodied agents, it is required that the model structure be optimized for specific tasks and operational conditions. Such optimization can be static such as model compression or dynamic such as adaptive inference. Yet, these techniques have not been fully investigated for embodied control systems subject to time constraints, which necessitate sequential decision-making for multiple tasks, each with distinct inference latency limitations. In this paper, we present MoDeC, a time constraint-aware embodied control framework using the modular model adaptation. We formulate model adaptation to varying operational conditions on resource and time restrictions as dynamic routing on a modular network, incorporating these conditions as part of multi-task objectives. Our evaluation across several vision-based embodied environments demonstrates the robustness of MoDeC, showing that it outperforms other model adaptation methods in both performance and adherence to constraints in robotic manipulation and autonomous driving applications.