With advances in mobile technologies, social networks and global positioning (GPS) in the digital world, alternative mobility systems (taxis, carpool, demand-responsive services, peer-to-peer ridesharing, carsharing) have garnered interest from both public and private sectors as potential solutions to address last mile problem in public transit. Although there are number of models to optimize flexible or dynamic transit operations there has not been any methodology to evaluate equilibrium demand and effect on social welfare for these systems in an integrated supply-demand context. This study lays the groundwork for studying the equilibrium of these systems, and proposes an agent-based adjustment process to evaluate the properties of a stable sate as an agent-based stochastic user equilibrium (SUE). Four sets of experiments are conducted: (1) illustration with a simple 2-link network, (2) evaluation of a dynamic dial-a-ride policy, and (3 &4) illustration using real data from Oakville, Ontario & Manhattan, NY. The experiments demonstrate that the proposed model with multiple sample populations can generate an invariant distribution of demand and welfare effects and it can effectively be used to measure the effect of changes in flexible transport services operation policies on ridership. Moreover, this study also explores flexible transport services as two-sided markets, and extends the proposed agent-based day-to-day adjustment process to include day-to-day adjustment of the service operator(s) as a two-sided market. Additional computational experiments and a case study are conducted. Findings confirm the existence of thresholds from which network externalities cause two-sided and one-sided market equilibria to diverge. The Ramsey pricing criterion is used for social optimum to show that perfectly matched states from the proposed day-to-day process are equivalent to a social optimum. A case study using real data from Oakville, Ontario, as a first/last mile problem example demonstrates the sensitivity of the two-sided day-to-day model to operating policies.