This paper considers identification and estimation in models imposing conditional independence restrictions and featuring a scalar disturbance. It is shown that for this class of models the disturbance is endowed with a specific structure that is highlighted and exploited to obtain full knowledge of the structural function. Structural effects of a policy or treatment are allowed to vary across subpopulations that can be located on the joint distribution of unobservables of the model. In nonseparable triangular models with continuous endogenous variables this approach delivers identification of structural functions conditional on values of the control variable. These results are obtained after a reanalysis of local identification in nonseparable triangular models, where the connection between Chesher (2003) and Imbens and Newey (2009) is made explicit. It is also shown that in the presence of nonmonotone continuous instruments, nonseparable triangular models are always overidentifying. A generic estimation framework is described, and an analog estimator based on a new regression method ("dual regression") is proposed. An empirical application illustrates the methodology by estimating gasoline demand functions in the United States.