Queueing problems arise commonly in transportation and healthcare systems. With the immense popularity of shared mobility in recent years, there has been an increase in research interests on studying how such systems work from a queueing perspective and how to design policies to improve the system performance. In this thesis, we construct different Markovian queueing models and derive limit theorems for their empirical processes to study large scale dynamics of bike sharing, e-scooter sharing and healthcare systems. We also explore how different factors impact their system performance. First, we propose a stochastic model for bike sharing system with finite station capacity and non-stationary arrivals. In addition, we examine the power of information by extending the model to incorporate choice modeling where customers have higher probability of going to stations with more bikes. Then, we study the power of patient flexibility in the healthcare setting, and propose a model where a fraction p of all patients are flexible via joining the shortest of d queues, while the remaining 1-p only join the queue of their choice. Finally, we propose a model for e-scooter sharing that captures the battery life dynamics of a large scooter network. We prove a series of mean field limits and central limit theorem results for the empirical measure process of these queueing models, which provide insights on the mean, variance, and sample path dynamics of such large scale systems. We also show that an interchanging of limits result holds for both the mean field limit and the diffusion limit for some models. The analysis presented in this thesis gives an analytical framework for providing estimations of important performance measures of these stochastic systems and their confidence intervals. It also helps quantify the impact of information and flexibility. All of the above has the potential to inform better operations and designs of future systems in shared mobility and healthcare.