Causal decomposition analyses can help build the evidence base for interventions that address health disparities (inequities). They ask how disparities in outcomes may change under hypothetical intervention. Through study design and assumptions, they can rule out alternate explanations such as confounding, selection-bias, and measurement error, thereby identifying potential targets for intervention. Unfortunately, the literature on causal decomposition analysis and related methods have largely ignored equity concerns that actual interventionists would respect, limiting their relevance and practical value. This paper addresses these concerns by explicitly considering what covariates the outcome disparity and hypothetical intervention adjust for (so-called allowable covariates) and the equity value judgements these choices convey, drawing from the bioethics, biostatistics, epidemiology, and health services research literatures. From this discussion, we generalize decomposition estimands and formulae to incorporate allowable covariate sets, to reflect equity choices, while still allowing for adjustment of non-allowable covariates needed to satisfy causal assumptions. For these general formulae, we provide weighting-based estimators based on adaptations of ratio-of-mediator-probability and inverse-odds-ratio weighting. We discuss when these estimators reduce to already used estimators under certain equity value judgements, and a novel adaptation under other judgements.Comment: 39 pages, 1 Table and 1 Figure in the main text, 1 Figure in the Supplement. Notational system changed for clarity