Monte Carlo rendering and modern generative models both transform uncertain states into structured images, yet they are usually studied as separate processes. We introduce Monte Carlo Transport Scheduling, a framework that treats progressive path tracing as a continuous sampling-driven transport process. Our key observation is that the renderer already produces physically valid states along this process: nested Monte Carlo estimates trace a refinement trajectory whose natural time coordinate follows from sampling variance. This view leads to a continuous training framework that learns from real render endpoints rather than synthetic interpolants, preserving the statistical structure of Monte Carlo estimation while enabling arbitrary-step neural refinement. We evaluate the framework on a controlled rendering benchmark designed to separate transport difficulty from scene context, and show that it yields stable render refinement, supports continuous stopping between rendering states, and transfers as a physical prior for frozen generative samplers. These results suggest a common continuous-time substrate for rendering and generation, where Monte Carlo sampling provides both the physical states and the supervision for learning image transport.