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With the justification of a comprehensive matching function approach, this study analyzes the ride-sourcing market with heterogeneous users. A single origin-destination (O-D) ride-sourcing market model is first developed. The model fills the major research gap to extend general user heterogeneity modeling from the transit market to the two-sided ride-sourcing market. Sufficient conditions that guarantee a unique market equilibrium are given. Equilibrium market operation with a profit-maximizing platform is explored. Equilibrium transitions with respect to the changes in different exogenous market variables are investigated. Nonequilibrium modeling to understand the transition path is also analyzed, revealing how the detailed transition evolves. Extensions of the single O-D market model to aggregate and disaggregate markets and incorporation of travel time reliability are briefly discussed. Numerical experiments, which are based on real-world data in the city of Ningbo, China, and cover an aggregate/single O-D base case, sensitivity analysis, equilibrium transitions, and disaggregate market equilibrium, are presented to illustrate the theoretical model. The larger network of Sioux Falls is also tested as a demonstration of the disaggregate market equilibrium. Discussions are made regarding operation and policy implications. [ABSTRACT FROM AUTHOR]