Kubernetes (k8s) has the potential to merge the distributed edge and the cloud but lacks a scheduling framework specifically for edge-cloud systems. Besides, the hierarchical distribution of heterogeneous resources and the complex dependencies among requests and resources make the modeling and scheduling of k8s-oriented edge-cloud systems particularly sophisticated. In this paper, we introduce KaiS, a learning-based scheduling framework for such edge-cloud systems to improve the long-term throughput rate of request processing. First, we design a coordinated multi-agent actor-critic algorithm to cater to decentralized request dispatch and dynamic dispatch spaces within the edge cluster. Second, for diverse system scales and structures, we use graph neural networks to embed system state information, and combine the embedding results with multiple policy networks to reduce the orchestration dimensionality by stepwise scheduling. Finally, we adopt a two-time-scale scheduling mechanism to harmonize request dispatch and service orchestration, and present the implementation design of deploying the above algorithms compatible with native k8s components. Experiments using real workload traces show that KaiS can successfully learn appropriate scheduling policies, irrespective of request arrival patterns and system scales. Moreover, KaiS can enhance the average system throughput rate by 14.3% while reducing scheduling cost by 34.7% compared to baselines.