Summary: We consider the problem of estimating the conditional distribution \(\operatorname{P}(Y\in A|X)\) of a functional data object \(Y=(Y(t):t\in [0,1])\) in the space of continuous functions, given covariates \(X\) in a general space and assuming that \(Y\) and \(X\) are related by a functional linear regression model. Two estimation methods are proposed, based on either the empirical distribution of the estimated model residuals, or fitting functional parametric models to the model residuals. We show that consistent estimation can be achieved under relatively mild assumptions. We exemplify a general class of sets \(A\) specifying path properties of \(Y\) that are of interest in applications. The proposed methods are studied in several simulation experiments, and data analyses of electricity price and pollution curves.