Despite the massive success of fine-tuning large Pre-trained Language Models (PLMs) on a wide range of Natural Language Processing (NLP) tasks, they remain susceptible to out-of-distribution (OOD) and adversarial inputs. Data map (DM) is a simple yet effective dual-model approach that enhances the robustness of fine-tuned PLMs, which involves fine-tuning a model on the original training set (i.e. reference model), selecting a specified fraction of important training examples according to the training dynamics of the reference model, and fine-tuning the same model on these selected examples (i.e. main model). However, it suffers from the drawback of requiring fine-tuning the same model twice, which is computationally expensive for large models. In this paper, we first show that 1) training dynamics are highly transferable across different model sizes and different pre-training methods, and that 2) main models fine-tuned using DM learn faster than when using conventional Empirical Risk Minimization (ERM). Building on these observations, we propose a novel fine-tuning approach based on the DM method: Fine-Tuning by transFerring Training dynamics (FTFT). Compared with DM, FTFT uses more efficient reference models and then fine-tunes more capable main models for fewer steps. Our experiments show that FTFT achieves better generalization robustness than ERM while spending less than half of the training cost.