Selective algorithmic auditing requires deciding which process evidence should receive attention when exhaustive review is infeasible. This Review introduces a four-layer framework that connects process representation, learning, inspection allocation, and governance within a single budgeted sequential decision problem over event streams. Unlike prior reviews centered on predictive process monitoring, explainability, cost analysis, or bibliometric structure, the proposed framework examines how these functions interact when human review, computation, latency, and documentation capacity are constrained. A structured and targeted survey of 89 unique publication families is used to illustrate and critically examine event-log, Petri-net, graph, object-centric, neural, uncertainty-aware, sequential, bandit, reinforcement learning, and audit architecture approaches. The reviewed evidence indicates that substantial bodies of work address the individual layers, but cross-layer evaluation remains fragmented and uses heterogeneous datasets, objectives, and validation protocols. The synthesis identifies five priorities: audit-ready benchmarks, explicit inspection budget protocols, calibrated uncertainty, transfer across organizational contexts, and reproducible governance interfaces. The main contribution is a computational framework and a corpus-bounded research agenda that connects representation, learning, inspection allocation, and governance for selective algorithmic auditing.