We develop the Markovian dynamic transit assignment (MDTrA) modeling framework for large multi-modal networks to incorporate into the transit assignment analysis the dynamic aspects that come from the time-dependent relationship between demand and supply and the stochasticity that comes from differences in passenger's costs perception, measurement errors, and other sources of uncertainty. The intuition is that passengers choose their routes by a recursive arc-choice process, according to the expected minimum costs from their current node to their destinations. Our approach presents an important opportunity to use smartcard data, as demand profiles and users' choices over time are obtained from it.