Belonging to the family of Bayesian nonparametrics, Gaussian process (GP) based approaches have well-documented merits not only in learning over a rich class of nonlinear functions, but also in quantifying the associated uncertainty. However, most GP methods rely on a single preselected kernel function, which may fall short in characterizing data samples that arrive sequentially in time-critical applications. To enable online kernel adaptation, the present work advocates an incremental ensemble (IE-) GP framework, where an EGP assembler employs an ensemble of GP learners, each having a unique kernel belonging to a prescribed kernel dictionary. With each GP expert leveraging the random feature-based approximation to perform online prediction and model update with scalability, the EGP assembler capitalizes on data-adaptive weights to synthesize the per-expert predictions. Further, the novel IE-GP is generalized to accommodate time-varying functions by modeling structured dynamics at the EGP assembler and within each GP learner. To benchmark the performance of IE-GP and its dynamic variant in the adversarial setting where the modeling assumptions are violated, rigorous performance analysis has been conducted via the notion of regret, as the norm in online convex optimization. Last but not the least, online unsupervised learning for dimensionality reduction is explored under the novel IE-GP framework. Synthetic and real data tests demonstrate the effectiveness of the proposed schemes.