The edge, the fog, the cloud, and even the end-user's devices play a key role in the management of the health sensitive content/data lifecycle. However, the creation and management of solutions including multiple applications executed by multiple users in multiple environments (edge, the fog, and the cloud) to process multiple health repositories that, at the same time, fulfilling non-functional requirements (NFRs) represents a complex challenge for health care organizations. This paper presents the design, development, and implementation of an architectural model to create, on-demand, edge-fog-cloud processing structures to continuously handle big health data and, at the same time, to execute services for fulfilling NFRs. In this model, constructive and modular blocks, implemented as microservices and nanoservices, are recursively interconnected to create edge-fog-cloud processing structures as infrastructureagnostic services. Continuity schemes create dataflows through the blocks of edge-fog-cloud structures and enforce, in an implicit manner, the fulfillment of NFRs for data arriving and departing to/from each block of each edge-fog-cloud structure. To show the feasibility of this model, a prototype was built using this model, which was evaluated in a case study based on the processing of health data for supporting critical decision-making procedures in remote patient monitoring. This study considered scenarios where end-users and medical staff received insights discovered when processing electrocardiograms (ECGs) produced by sensors in wireless IoT devices as well as where physicians received patient records (spirometry studies, ECGs and tomography images) and warnings raised when online analyzing and identifying anomalies in the analyzed ECG data. A scenario where organizations manage multiple simultaneous each edge-fog-cloud structure for processing of health data and contents delivered to internal and external staff was also studied. The evaluation of these scenarios showed the feasibility of applying this model to the building of solutions interconnecting multiple services/applications managing big health data through different environments. INDEX TERMS Big health data, edge-fog-cloud, health-IoT processing, Internet of Things, microservice architecture.