Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks. We consider the transductive setting, where decisions are made on a test sample of $m$ new points, giving rise to $m$ conformal $p$-values. {While classical results only concern their marginal distribution, we show that their joint distribution follows a P\'olya urn model, and establish a concentration inequality for their empirical distribution function.} The results hold for arbitrary exchangeable scores, including {\it adaptive} ones that can use the covariates of the test+calibration samples at training stage for increased accuracy. We demonstrate the usefulness of these theoretical results through uniform, in-probability guarantees for two machine learning tasks of current interest: interval prediction for transductive transfer learning and novelty detection based on two-class classification.

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