The study of semantic shifts, that is, of how words change meaning as a consequence of social practices, events and political circumstances, is relevant in Natural Language Processing, Linguistics, and Social Sciences. The increasing availability of large diachronic corpora and advance in computational semantics have accelerated the development of computational approaches to detecting such shifts. In this paper, we introduce a novel approach to tracing the evolution of word meaning over time. Our analysis focuses on gradual changes in word semantics and relies on an incremental approach to semantic shift detection (SSD) called WiDiD. WiDiD leverages scalable and evolutionary clustering of contextualised word embeddings to detect semantic shifts and capture temporal transactions in word meanings. Existing approaches to SSD (a) significantly simplify the semantic shift problem to cover change between two (or a few) time points, and (b) consider the existing corpora as static. We instead treat SSD as an organic process in which word meanings evolve across tens or even hundreds of time periods as the corpus is progressively made available. This results in an extremely demanding task that entails a multitude of intricate decisions. We demonstrate the applicability of this incremental approach on a diachronic corpus of Italian parliamentary speeches spanning eighteen distinct time periods. We also evaluate its performance on seven popular labelled benchmarks for SSD across multiple languages. Empirical results show that our results are at least comparable to state-of-the-art approaches, while outperforming the state-of-the-art for certain languages.