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Emergent constraints are quantities that are observable from current measurements and have skill predicting future climate. This study explores 19 previously proposed emergent constraints related to equilibrium climate sensitivity (ECS; the global-average equilibrium surface temperature response to CO2 doubling). Several constraints are shown to be closely related, emphasizing the importance for careful understanding of proposed constraints. A new method is presented for decomposing correlation between an emergent constraint and ECS into terms related to physical processes and geographical regions. Using this decomposition, one can determine whether the processes and regions explaining correlation with ECS correspond to the physical explanation offered for the constraint. Shortwave cloud feedback is generally found to be the dominant contributor to correlations with ECS because it is the largest source of intermodel spread in ECS. In all cases, correlation results from interaction between a variety of terms, reflecting the complex nature of ECS and the fact that feedback terms and forcing are themselves correlated with each other. For 4 of the 19 constraints, the originally proposed explanation for correlation is borne out by our analysis. These four constraints all predict relatively high climate sensitivity. The credibility of six other constraints is called into question owing to correlation with ECS coming mainly from unexpected sources and/or lack of robustness to changes in ensembles. Another six constraints lack a testable explanation and hence cannot be confirmed. The fact that this study casts doubt upon more constraints than it confirms highlights the need for caution when identifying emergent constraints from small ensembles.