Many high-throughput scientific experiments produce distributional outputs, requiring conditional generative models to interpret the data, and to predict outcomes at new inputs. Acquiring experimental data at new conditions (e.g., for a new antigen or a new cell type) is often expensive, motivating the need for automated experimental design methods. However, existing active learning frameworks are limited as they assume scalar or vector-valued outputs. We first show that the Wasserstein test risk of a conditional generative task decomposes into a mean term and a shape term, demonstrating why mean-based active learning is blind to distributional shape. To capture both, we propose the transport Neural Tangent Kernel (tNTK), a tractable kernel that measures how sensitive a generative model's predicted distribution is with respect to its parameters. The tNTK recovers the standard NTK in the deterministic limit and upper-bounds the posterior Fréchet variance of the prediction under small parameter perturbations. It admits a closed-form expression for affine Gaussian transport and tractable Monte Carlo estimators otherwise. Treating the tNTK as the covariance function of a Gaussian process, we greedily select inputs that minimize the total GP posterior variance across the pool, yielding a distribution-aware active learning strategy. We evaluate this strategy across four classes of conditional generative models, including conditional flow matching and conditional Monge gap, on synthetic distributions, intent-labeled sentences, and single-cell perturbation responses. In these experiments, the tNTK outperforms the baseline NTK computed from the conditional mean, demonstrating the value of distribution-aware active learning for generative-modeling tasks.

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