With advancements in domain generalized stereo matching networks, models pre-trained on synthetic data demonstrate strong robustness to unseen domains. However, few studies have investigated the robustness after fine-tuning them in real-world scenarios, during which the domain generalization ability can be seriously degraded. In this paper, we explore fine-tuning stereo matching networks without compromising their robustness to unseen domains. Our motivation stems from comparing Ground Truth (GT) versus Pseudo Label (PL) for fine-tuning: GT degrades, but PL preserves the domain generalization ability. Empirically, we find the difference between GT and PL implies valuable information that can regularize networks during fine-tuning. We also propose a framework to utilize this difference for fine-tuning, consisting of a frozen Teacher, an exponential moving average (EMA) Teacher, and a Student network. The core idea is to utilize the EMA Teacher to measure what the Student has learned and dynamically improve GT and PL for fine-tuning. We integrate our framework with state-of-the-art networks and evaluate its effectiveness on several real-world datasets. Extensive experiments show that our method effectively preserves the domain generalization ability during fine-tuning. Code is available at: https://github.com/jiaw-z/DKT-Stereo.