A growing number of studies identify a generalized slowdown in labor productivity growth. The very existence of the slowdown ignited a series of academic debates suggesting that secular stagnation or "mismeasurement" problems are at the root of the observed trends. We posit that the composition of aggregate productivity matters. In a nutshell, we make the analysis of productivity growth slowdown more fine-grained by shifting the focus to the industry level, considering that the downward trend identified at the macroeconomic level emerges from the aggregation of diverse industry-level productivity trends. We perform an analysis of the structural dynamics of labor productivity by conducting a non-parametric dynamic decomposition exercise that separates within (improvement) and between (structural change) effects for 10 OECD countries. By pooling industries in groups identified according to two different taxonomies - one related to R&D intensities rankings, and the other built upon the Pavitt taxonomy of sources of technological change -, this study assess the industry-level contributions to the slowdown and the trends over time of the within and between components. We interpret our findings highlighting common patterns and suggest two related technological explanations for the productivity slowdown: one based on a Baumol-disease-like effect driven by structural change and another based on implementation lags and/or on an exhaustion of technological opportunities - that is, on decreasing returns in innovative activities. To investigate that, we complement our productivity analysis with evidence on innovation slowdown trends, looking at aggregate and compositional trends. We explore the innovation slowdown using an array of indicators based on the notion of "idea-TFP" and show that there is a generalized evidence for its occurrence. Eventually, we relate productivity and innovation slowdowns deriving tables of trends co-movements, weighted by input-output matrices coefficients, and clustered by Pavitt industry group. We interpret these relationships and highlight patterns and clusters of significant correlations.