Recent results show that several important graph classes can be embedded as subgraphs of strong products of simpler graphs classes (paths, small cliques, or graphs of bounded treewidth). This paper develops general techniques to bound the chromatic number (and its popular variants, such as fractional, clustered, or defective chromatic number) of the strong product of general graphs with simpler graphs classes, such as paths, and more generally graphs of bounded treewidth. We also highlight important links between the study of (fractional) clustered colouring of strong products and other topics, such as asymptotic dimension in metric theory and topology, site percolation in probability theory, and the Shannon capacity in information theory.