This work explores the integration and experimental evaluation of Fluid Computing principles with the Internet of Things (IoT) through the concept of Fluid Digital Twins (FDTs). They have been recently introduced as a cyber-physical paradigm designed to serve as intermediate software components aiming to enable seamless task migration, optimize resource utilization, and streamline interactions. Expanding upon this investigation, the research investigates FDTs within the context of the edge-to-cloud compute continuum. It models and explores the feasibility and ramifications of deploying and orchestrating FDTs and their dynamic capabilities across diverse computational facilities, from edge devices to cloud infrastructure. The paper outlines a new distributed FDT's modeling, presents the implemented prototype within a target reference use case together with its experimental evaluations, and analyzes challenges and opportunities inherent in this dynamic integration.