In this paper, we address the Online Unsupervised Domain Adaptation (OUDA) problem and propose a novel multi-stage framework to solve real-world situations when the target data are unlabeled and arriving online sequentially in batches. Most of the traditional manifold-based methods on the OUDA problem focus on transforming each arriving target data to the source domain without sufficiently considering the temporal coherency and accumulative statistics among the arriving target data. In order to project the data from the source and the target domains to a common subspace and manipulate the projected data in real-time, our proposed framework institutes a novel method, called an Incremental Computation of Mean-Subspace (ICMS) technique, which computes an approximation of mean-target subspace on a Grassmann manifold and is proven to be a close approximate to the Karcher mean. Furthermore, the transformation matrix computed from the mean-target subspace is applied to the next target data in the recursive-feedback stage, aligning the target data closer to the source domain. The computation of transformation matrix and the prediction of next-target subspace leverage the performance of the recursive-feedback stage by considering the cumulative temporal dependency among the flow of the target subspace on the Grassmann manifold. The labels of the transformed target data are predicted by the pre-trained source classifier, then the classifier is updated by the transformed data and predicted labels. Extensive experiments on six datasets were conducted to investigate in depth the effect and contribution of each stage in our proposed framework and its performance over previous approaches in terms of classification accuracy and computational speed. In addition, the experiments on traditional manifold-based learning models and neural-network-based learning models demonstrated the applicability of our proposed framework for various types of learning models.