In this work, we address the important question of adaptability of artificial neural networks (NNs) used for impairment mitigation in optical transmission systems. We demonstrate that by using well-developed techniques based on the concept of transfer learning, we can efficaciously retrain NN-based equalizers to adapt changes in the transmission system using just a fraction of the initial training data and epochs. We evaluate the potential of transfer learning to adapt the NN to changes in the launch powers, modulation formats, symbol rates, or even fiber plants (different fiber types and lengths). The numerical examples utilize the recently introduced NN equalizer consisting of a convolutional layer coupled with bi-directional long-short term memory (biLSTM) recurrent NN element. Our analysis focuses on long-haul coherent optical transmission systems for two types of fibers: the standard single-mode fiber (SSMF) and the TrueWave Classic (TWC) fiber. We underline the specific peculiarities that occur when transferring the learning in coherent optical communication systems and draw the limits for the transfer learning efficiency. Our results demonstrate the effectiveness of transfer learning for the fast adaptation of NN architectures to different transmission regimes and scenarios, paving the way for engineering flexible and universal solutions for nonlinearity mitigation.