Pedestrian trajectory prediction is crucial for many intelligent tasks. While existing methods predict future trajectories from fixed-frame historical observations, they are limited by the observational perspective and the need for extensive historical information, resulting in prediction delays and inflexible generalization in real-time systems. In this paper, we propose a novel task called Transferable Online Pedestrian Trajectory Prediction (TOTP), which synchronously predicts future trajectories with variable observations and enables effective task transfer under different observation constraints. To advance TOTP modeling, we propose a Temporal-Adaptive Mamba Latent Diffusion (TAMLD) model. It utilizes the Social-Implicit Mamba Synthesizer to extract motion states with social interaction and refine temporal representations through Temporal-Aware Distillation. A Trend-Conditional Mamba Decomposer generates the motion latent distribution of the future motion trends and predicts future motion trajectories through sampling decomposition. We utilize Motion-Latent Mamba Diffusion to reconstruct the latent space disturbed by imbalanced temporal noise. Our method achieves state-of-the-art results on multiple datasets and tasks, showcasing temporal adaptability and strong generalization.