Classifying moral values in user-generated text from social media is critical in understanding community cultures and interpreting user behaviors of social movements. Moral values and language usage can change across the social movements; however, text classifiers are usually trained in source domains of existing social movements and tested in target domains of new social issues without considering the variations. In this study, we examine domain shifts of moral values and language usage, quantify the effects of domain shifts on the morality classification task, and propose a neural adaptation framework via instance weighting to improve cross-domain classification tasks. The quantification analysis suggests a strong correlation between morality shifts, language usage, and classification performance. We evaluate the neural adaptation framework on a public Twitter data across 7 social movements and gain classification improvements up to 12.1\%. Finally, we release a new data of the COVID-19 vaccine labeled with moral values and evaluate our approach on the new target domain. For the case study of the COVID-19 vaccine, our adaptation framework achieves up to 5.26\% improvements over neural baselines.