For the past 40 years, cellular industry has been relying on static radio access deployments with gross over-provisioning. However, to meet the exponentially growing volumes of irregular data, the very notion of a cell will have to be rethought to allow them be (re-)configured on-demand and in automated manner. This work puts forward a vision of moving networks to match dynamic user demand with network access supply in the beyond-5G cellular systems. The resulting adaptive and flexible network infrastructures will leverage intelligent capable devices (e.g., cars and drones) by employing appropriate user involvement schemes. This work is a recollection of our efforts in this space with the goal to contribute a comprehensive research agenda. Particular attention is paid to quantifying the network performance scaling and session continuity gains with ultra-dense moving cells. Our findings argue for non-incremental benefits of integrating moving access points on a par with conventional (static) cellular access infrastructure.