According to World Health Organization (WHO), the worldwide prevalence of chronic diseases increases fast and new threats, such as Covid-19 pandemic, continue to emerge, while the aging population continues decaying the dependency ratio. These challenges will cause a huge pressure on the efficacy and cost-efficiency of healthcare systems worldwide. Thanks to the emerging technologies, such as novel medical imaging and monitoring instrumentation, and Internet of Medical Things (IoMT), more accurate and versatile patient data than ever is available for medical use. To transform the technology advancements into better outcome and improved efficiency of healthcare, seamless interoperation of the underlying key technologies needs to be ensured. Novel IoT and communication technologies, edge computing and virtualization have a major role in this transformation. In this article, we explore the combined use of these technologies for managing complex tasks of connecting patients, personnel, hospital systems, electronic health records and medical instrumentation. We summarize our joint effort of four recent scientific articles that together demonstrate the potential of the edge-cloud continuum as the base approach for providing efficient and secure distributed e-health and e-welfare services. Finally, we provide an outlook for future research needs.