Modern socio-technical systems, such as smart energy grids, ride-hailing services, or digital marketplaces, typically consist of many interconnected users and competing service providers. Within these systems, notions like market equilibrium are tightly connected to the ``evolution'' of the network of users. In this paper, we model the users' state and dynamics as a linear dynamical system, and the service providers as agents taking part to a generalized Nash game, whose outcome coincides with the input of the users' dynamics. We are thus able to characterize the notion of co-evolution of the market and the network dynamics and derive conditions leading to a pertinent notion of equilibrium. These conditions are based on dissipativity arguments and yield easy-to-check linear matrix inequalities. We then turn the problem into a control one: how can we incentivize or penalize the service providers acting as little as possible to steer the whole network to a desirable outcome? This so-called light-touch policy design problem can be solved through bilinear matrix inequalities. We also provide a dimensionality-reduction procedure, which offers network-size independent conditions and design tools. Finally, we illustrate our novel notions and algorithms on a simulation setup stemming from digital market regulations for influencers, a topic of growing interest.