Knowledge Graphs (KGs) are foundational to applications such as search, question answering, and recommendation. Conventional knowledge graph construction methods are predominantly static, relying on a single-step construction from a fixed corpus with a predefined schema. However, such methods are suboptimal for real-world scenarios where data arrives dynamically, as incorporating new information requires complete and computationally expensive graph reconstructions. Furthermore, predefined schemas hinder the flexibility of knowledge graph construction. To address these limitations, we introduce DIAL-KG, a closed-loop framework for incremental KG construction orchestrated by a Meta-Knowledge Base (MKB). The framework operates in a three-stage cycle: (i) Dual-Track Extraction, which ensures knowledge completeness by defaulting to triple generation and switching to event extraction for complex knowledge; (ii) Governance Adjudication, which ensures the fidelity and currency of extracted facts to prevent hallucinations and knowledge staleness; and (iii) Schema Evolution, in which new schemas are induced from validated knowledge to guide subsequent construction cycles, and knowledge from the current round is incrementally applied to the existing KG. Extensive experiments demonstrate that our framework achieves state-of-the-art (SOTA) performance in the quality of both the constructed graph and the induced schemas.