Text-driven video editing aims to modify video content based on natural language instructions. While recent training-free methods have leveraged pretrained diffusion models, they often rely on an inversion-editing paradigm. This paradigm maps the video to a latent space before editing. However, the inversion process is not perfectly accurate, often compromising appearance fidelity and motion consistency. To address this, we introduce FlowDirector, a novel training-free and inversion-free video editing framework. Our framework models the editing process as a direct evolution in the data space. It guides the video to transition smoothly along its inherent spatio-temporal manifold using an ordinary differential equation (ODE), thereby avoiding the inaccurate inversion step. From this foundation, we introduce three flow correction strategies for appearance, motion, and stability: 1) Direction-aware flow correction amplifies components that oppose the source direction and removes irrelevant terms, breaking conservative streamlines and enabling stronger structural and textural changes. 2) Motion-appearance decoupling optimizes motion agreement as an energy term at each timestep, significantly improving consistency and motion transfer. 3) Differential averaging guidance strategy leverages differences among multiple candidate flows to approximate a low variance regime at low cost, suppressing artifacts and stabilizing the trajectory. Extensive experiments across various editing tasks and benchmarks demonstrate that FlowDirector achieves state-of-the-art performance in instruction following, temporal consistency, and background preservation, establishing an efficient new paradigm for coherent video editing without inversion.