Uniform marginal prices cannot generally support competitive equilibrium solutions in markets with non-convexities and yield schedules' inconsistency and surplus sub-optimality effects. Previous work has identified non-convexities associated with the generation side of electricity markets and proposed a generalized uplift approach to eliminate these effects. This paper examines the above issues from the perspective of the flexible demand (FD) side. FD non-convexities are identified, including its ability to forgo demand activities and minimum power levels, and resulting inconsistency and surplus sub-optimality effects are demonstrated through simple examples. Generalized uplift functions for FD participants are proposed, including quadratic pricing terms to limit their tendency to concentrate at the lowest-priced periods, and binary terms associated with their ability to forgo activities. Finally, a new rule is proposed for the equitable distribution of the total surplus loss among the market participants. These contributions are supported by case studies on a market with a day-ahead horizon and hourly resolution.