Building Free-Viewpoint Videos in a streaming manner offers the advantage of rapid responsiveness compared to offline training methods, greatly enhancing user experience. However, current streaming approaches face challenges of high per-frame reconstruction time (10s+) and error accumulation, limiting their broader application. In this paper, we propose Instant Gaussian Stream (IGS), a fast and generalizable streaming framework, to address these issues. First, we introduce a generalized Anchor-driven Gaussian Motion Network, which projects multi-view 2D motion features into 3D space, using anchor points to drive the motion of all Gaussians. This generalized Network generates the motion of Gaussians for each target frame in the time required for a single inference. Second, we propose a Key-frame-guided Streaming Strategy that refines each key frame, enabling accurate reconstruction of temporally complex scenes while mitigating error accumulation. We conducted extensive in-domain and cross-domain evaluations, demonstrating that our approach can achieve streaming with a average per-frame reconstruction time of 2s+, alongside a enhancement in view synthesis quality.