Studying migration patterns driven by extreme environmental events is crucial for building a sustainable society and stable economy. Motivated by a real dataset about human migrations, this paper develops a transformed varying coefficient model for origin and destination (OD) regression to elucidate the complex associations of migration patterns with spatio-temporal dependencies and socioeconomic factors. Existing studies often overlook the dynamic effects of these factors in OD regression. Furthermore, with the increasing ease of collecting OD data, the scale of current OD regression data is typically large, necessitating the development of methods for efficiently fitting large-scale migration data. We address the challenge by proposing a new Bayesian interpretation for the proposed OD models, leveraging sufficient statistics for efficient big data computation. Our method, inspired by migration studies, promises broad applicability across various fields, contributing to refined statistical analysis techniques. Extensive numerical studies are provided, and insights from real data analysis are shared.