Research Infrastructures (RIs) are strategic assets facilitating innovation and knowledge advancement across all scientific disciplines. They provide researchers with advanced tools and resources that go beyond individual or institutional capacities and promote collaboration, community-building and the application of scientific standards. Remote and virtual access to RIs enables scientists to use these essential resources without the necessity of being physically present. The COVID-19 pandemic restrictions where a catalyst for the expansion and further development of remote and virtual access models, particularly in fields where physical access had been the predominant model. The eRImote project explores pathways for digital and remote RI access through targeted surveys, stakeholder workshops, expert groups discussions, and the analysis of specific use cases. This paper provides a definition of remote and virtual access and remote training and explores their implementation across various RIs, highlighting the implications for their operational processes and the dynamics of interaction between RIs and their user communities. It presents the identified advantages, obstacles, and best-practices, alongside strategies and recommendations to navigate and mitigate challenges effectively. Key issues and recommendations are summed up separately for remote access, virtual access, and remote training, complemented by general recommendations for facilitating remote and virtual access to RIs. These relate to budgeting and funding, the balancing of RI access models, the need for regulatory frameworks for sample shipments, collaboration among RIs, impact assessment of remote and virtual access on user interactions, operational efficiency and the environment footprint of RIs, and the adaption of data sharing policies. Stakeholders are broadly invited to give their feedback on the paper's findings and conclusions, which will be integrated into improved versions of this paper.