Methods from machine learning (ML) have transformed the implementation of Perception-Cognition-Communication-Action loops in Cyber-Physical Systems (CPS) and the Internet of Things (IoT), replacing mechanistic and basic statistical models with those derived from data. However, the first generation of ML approaches, which depend on supervised learning with annotated data to create task-specific models, faces significant limitations in scaling to the diverse sensor modalities, deployment configurations, application tasks, and operating dynamics characterizing real-world CPS-IoT systems. The success of task-agnostic foundation models (FMs), including multimodal large language models (LLMs), in addressing similar challenges across natural language, computer vision, and human speech has generated considerable enthusiasm for and exploration of FMs and LLMs as flexible building blocks in CPS-IoT analytics pipelines, promising to reduce the need for costly task-specific engineering. Nonetheless, a significant gap persists between the current capabilities of FMs and LLMs in the CPS-IoT domain and the requirements they must meet to be viable for CPS-IoT applications. In this paper, we analyze and characterize this gap through a thorough examination of the state of the art and our research, which extends beyond it in various dimensions. Based on the results of our analysis and research, we identify essential desiderata that CPS-IoT domain-specific FMs and LLMs must satisfy to bridge this gap. We also propose actions by CPS-IoT researchers to collaborate in developing key community resources necessary for establishing FMs and LLMs as foundational tools for the next generation of CPS-IoT systems.