In this work, a new time-sequential positioning and fault detection method is derived and analyzed for dual-frequency, multi-constellation Advanced Receiver Autonomous Integrity Monitoring (ARAIM). Unlike conventional 'snapshot' ARAIM, the time-sequential approach exploits changes in satellite geometry at the cost of higher computation and memory loads. From the perspective of a user on earth, GNSS satellite motion is small over less-than-ten-minute-long time intervals. But, the accumulated geometry variations of redundant satellites from multiple GNSS can be substantial. This paper quantifies the potential performance benefit brought by satellite motion to ARAIM. The first research challenge is to define raw GNSS code and carrier error models, which must account for measurement error correlation over time. These models must also be consistent with currently established ARAIM assumptions on carrier-smoothed code. The second step is to use these raw measurements in estimators and fault-detectors capable of exploiting geometric diversity for positioning, float carrier phase cycle ambiguity estimation, and integrity risk evaluation. In this example implementation, signals from multiple GNSS are arranged in a finite-interval batch-type estimator, and sequentially processed in a sliding window mechanism We derive a compact, computationally-efficient carrier-smoothed-code-based batch implementation. Fault detection is carried out using a multiple hypothesis batch-solution separation algorithm. Availability is analyzed worldwide for aircraft precision approach navigation applications. Results show dramatic performance improvements for batch 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.