In this paper, we study a class of constrained mean-field-type games with several decision-makers interacting in a stationary environment. The payoff functionals depend not only on the strategy profile but also on the first moment. This novel static game approach leads to risk-aware solution concepts. We introduce several solution concepts: mean-field-type best-response strategies, mean-field-type generalized Nash equilibria, mean-field-type variational equilibria. We provide a structure of payoffs with a decomposition that includes variance-aware cost. We discuss some learning algorithms to reach mean-field-type best Nash equilibria under constraints. The methodology is extended to include migration costs and migration incentives for each decision-maker and each choice component.