The seamless integration of multiple radio access technologies (multi-RAT) and cloud/edge resources is pivotal for advancing future networks, which seek to unify distributed and heterogeneous computing and communication resources into a cohesive continuum system, tailored for mobile applications. Many research projects and focused studies are proposing solutions in this area, the impact of which is undoubtedly increased by moving from theoretical and simulation studies to experimental validations. To this aim, this paper proposes a testbed architecture that combines contemporary communication and cloud technologies to provide microservice-based mobile applications with the ability to offload part of their tasks to cloud/edge data centers connected by multi-RAT cellular networks. The testbed leverages Kubernetes, Istio service mesh, OpenFlow, public 5G networks, and IEEE 802.11ad mmWave (60 GHz) Wi-Fi access points. The architecture is validated through a use case in which a ground robot autonomously follows a moving object by using an artificial intelligence-driven computer vision application. Computationally intensive navigation tasks are offloaded by the robot to microservice instances, which are executed on demand within cloud and edge data centers that the robot can exploit during its journey. The proposed testbed is flexible and can be reused to assess communication and cloud innovations focusing on multi-RAT cloud continuum scenarios.