Recent years witnessed a rising interest towards Datalog-based ontological reasoning systems, both in academia and industry. These systems adopt languages, often shared under the collective name of Datalog$^\pm$, that extend Datalog with the essential feature of existential quantification, while introducing syntactic limitations to sustain reasoning decidability and achieve a good trade-off between expressive power and computational complexity. From an implementation perspective, modern reasoners borrow the vast experience of the database community in developing streaming-based data processing systems, such as volcano-iterator architectures, that sustain a limited memory footprint and good scalability. In this paper, we focus on two extremely promising, expressive, and tractable languages, namely, Shy and Warded Datalog$^\pm$. We leverage their theoretical underpinnings to introduce novel reasoning techniques, technically, ``chase variants'', that are particularly fit for efficient reasoning in streaming-based architectures. We then implement them in Vadalog, our reference streaming-based engine, to efficiently solve ontological reasoning tasks over real-world settings.