As large language models (LLMs) are increasingly adapted from general-purpose systems into domain-specific problem solvers, a central challenge is how to provide reliable process-level guidance for knowledge-intensive reasoning. High-quality reasoning supervision, such as annotated reasoning trajectories or externally generated rationales, is costly to obtain, difficult to verify, and hard to scale across domains. In this paper, we propose KG-RLVR, a lightweight framework that reframes KG-enhanced LLM post-training as verifiable reward construction from traceable structured evidence. KG-RLVR retrieves evidence from external knowledge graphs, consolidates noisy graph neighborhoods into representative evidence units, and converts these units into locally computable process rewards. The reward encourages the model to cover directly relevant evidence while selectively integrating less direct but potentially useful auxiliary evidence, thereby guiding reasoning traces without requiring externally authored process demonstrations. Experiments on knowledge-intensive medical and multi-domain QA benchmarks show that KG-RLVR consistently improves over vanilla RL, demonstrating that KG-derived process rewards provide practical and scalable supervision for reasoning-oriented LLM post-training.