This study considers a wireless network where an access point (AP) broadcasts timely updates to numerous mobile users. The timeliness of information owned by a user is characterized by the age of information. Frequently broadcasting the timely updates at constant maximum power can minimize the age of information for all users, but wastes valuable communication resources (ie., time and energy). To address the age-energy trade-off, it is critical to develop an efficient scheduling algorithm that identifies broadcast times and allocates power. Moreover, unpredictable user movement would cause rapidly varying communication channels; in particular, those channels can be non-stationary. Our main contribution is to develop an online scheduling algorithm and a channel-agnostic scheduling algorithm for such a mobile network with a provable performance guarantee.