Conformal prediction provides a distribution-free framework for uncertainty quantification via prediction sets with exact finite-sample coverage. In low dimensions these sets are easy to interpret, but in high-dimensional or structured output spaces they are difficult to represent and use, which can limit their ability to integrate with downstream tasks such as sampling and probabilistic forecasting. We show that any sufficiently regular differentiable nonconformity score induces a deterministic flow on the output space whose trajectories converge to the boundary of the corresponding conformal prediction set. This leads to a computationally efficient, training-free method for sampling conformal boundaries in arbitrary dimensions. Mixing across confidence levels yields conformal predictive distributions whose quantile regions coincide with the empirical conformal prediction sets. We provide an approximation bound decomposing CPD predictive error into score-induced distortion, base-measure quality, and gradient flow-induced distortion. We evaluate the approach on PDE inverse problems, precipitation downscaling, climate model debiasing, and hurricane trajectory forecasting.