The applications of dynamic content updates for a group of users, for example weather report and traffic congestion broadcast, have been shown to benefit significantly from Delay Tolerant Networks (DTN) communication mechanisms such as opportunistic contact and user sharing. In this paper, we study the dynamic content updates over energy-constrained DTN, and focus on its asymptotic properties about stability and scalability on the time and network size dimensions respectively. By introducing a continuous time Markov framework, we characterize the content dissemination system by both the source update & node sharing process and energy constraint. Based on this framework, we prove the system is stable when the time t approaches to infty, and the convergence rate is exponential. At the same time, by simulation we find that the content's average stable age decreases as Θ(√N) as the number of users N increases. Moreover, through extensive numerical results, we demonstrate the correctness of the theoretically obtained properties of stability.