In this work, a new time-sequential positioning and fault detection method is derived and analyzed for dualfrequency, multi-constellation Advanced Receiver Autonomous Integrity Monitoring (ARAIM). Unlike conventional 'snapshot' ARAIM, the sequential approach exploits changes in satellite geometry at the cost of slightly higher computation and memory loads. From the perspective of users on earth, the motion of any given GNSS satellite is small over short time intervals. But, the accumulated effect geometry variations of redundant satellites from multiple GNSS constellations can be substantial. This paper quantifies the potential performance benefit brought by satellite motion to ARAIM. It specifically addresses the following research challenges: (a) defining and experimentally validating raw GNSS code and carrier error models over time, consistent with established ARAIM assumptions, (b) designing estimators and fault-detectors capable of exploiting satellite motion for positioning, carrier phase cycle ambiguity estimation, and integrity evaluation, and (c) formulating these processes in a computationally-efficient implementation. A modular algorithm is designed, only requiring a minor augmentation of the snapshot airborne ARAIM multiple hypothesis solution separation (MHSS) algorithm. Other modifications to enable time-sequential ARAIM include additional ground segment performance commitments, and the inclusion of extra parameters in the broadcast integrity support message (ISM). Availability is analyzed worldwide for aircraft precision approach navigation applications. Results show substantial performance improvements for sequential ARAIM over snapshot ARAIM, not only to achieve 'localizer precision vertical' (LPV) requirements using depleted GPS and Galileo constellations, but also to fulfill much more stringent requirements including a ten-meter vertical alert limit.